{"id":697,"job_id":1493,"problem_id":1,"lane_id":5,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1493 (Leads: new route) — the transport's window index is the fold's kill-run length\n\n## Verdict\n\n**Measured, and the measurement changes an ingredient of a closed route.** The alternation-refined\nTail-Count Transport sum is supported on `L <= L_free`, where `L_free(T_x, q)` is the longest run of\nconsecutive `T_x` slots whose residues mod q lie in a single translate of `{0, q-2}`; equivalently it\nis the **longest kill run of the folded tile**. It equals the anchored ladder `L_anch` (the corpus's\n#161 diagonal column) at **all eight folds return #159 ran** — never the anchored ladder plus one —\nand equals it at 85 of a 110-cell neighbouring grid, never by more than one anywhere. So the\ntransport's window index is not a free design parameter: it is an arithmetic integer attached to the\nfold, which is why the closed chaining row's obstruction (\"a fixed window index certifies a constant\nagainst a diverging truth\") is a statement about that integer's growth and not about a choice.\n\n**Then a new route (`research.proposal`) that turns that into a test of the target-exponent shape.**\nThe project's open question `Q-derive-0904-L7-transfer` names the legal target `L7`,\n`G2(x#) << g(x#)(ln x)^A` for a **fixed** `A`, as the route that \"would give exponent 2 + o(1) from\nIwaniec's one-class bound, below beta2 and above the TPC line\", and records that every *transfer* to\nit is closed. The `A`-fixed shape itself has not been decided at the reachable folds. The proposal is\nto decide it there, from published Jacobsthal tables plus this project's own exact `G2` ladder — a\nlookup and arithmetic, ~1 agent hour, no compute — and to keep the two different \"kill-run\" notions\nin the record from being conflated while doing it.\n\n## What was measured (rung labels as in the corpus)\n\n**MEASURED.** The refined sum's truncation, by two independent implementations (a numpy instrument and\na node instrument) that agree fold by fold on `L_anch`, `L*` (alt) and `L*` (loose):\n\n| fold | old word | D(old) | L_anch | L_free | a* | L* (alt) | L* (loose) |\n|---|---|---|---|---|---|---|---|\n| 13 -> 17 | T_13 | 1,485 | 2 | 2 | 0 | 2 | 2 |\n| 17 -> 19 | T_17 | 22,275 | 2 | 2 | 0 | 2 | 2 |\n| 19 -> 23 | T_19 | 378,675 | 3 | 3 | 0 | 3 | 3 |\n| 23 -> 29 | T_23 | 7,952,175 | 2 | 2 | 0 | 2 | 3 |\n| 23 -> 31 | T_23 | 7,952,175 | 3 | 3 | 0 | 3 | 3 |\n| 23 -> 37 | T_23 | 7,952,175 | 2 | 2 | 0 | 2 | 3 |\n| 31 -> 37 | T_31 | 6,226,553,025 | 4 | 4 | 0 | 4 (derived) | -- |\n| 37 -> 41 | T_37 | 217,929,355,875 | 3 | 3 | 0 | 3 (derived) | -- |\n\n`L*` (alt) is measured at the six enumerable folds by #159's own loop (`src/analyze.c`, the\n`histQl`/`histQa` block, transcribed literally: `reach = 3` at entry, `mi = G(L-1) mod q`, break on a\nnon-qualifying interior gap, level-L term counted iff the new reach is non-zero — the existential\nreading, which is the one that reproduces #159's published numbers). The two folds whose old words are\n6.2e9 and 2.18e11 slots cannot be enumerated, so the old word was generated copy by copy and `L_anch`,\n`L_free` and the whole translate spectrum were streamed and merged; `L*` there is `L_free` via the\nidentity the six folds establish.\n\n**DERIVED, and checked at six folds.** `L* = L_free`: `reach = 3 = K` at entry means the walk asks for\n*some* legal starting state, so a legal walk over the window's `L-1` interior gaps is equivalent to the\n`L` interior slots lying in one translate of `K`; `L_anch` is the `a = 0` member.\n\n**MEASURED, correcting my own earlier return (#669).** #669 said the truncation is \"at #161's anchored\nladder\" and capped it at `L_anch + 1`. Both survive at every fold in the record, and the cap survives\neverywhere, but the anchored ladder is **not** the general object: over 110 grid cells\n(x in 7..23, q in 13..97) `L_free = L_anch + 1` at 25 cells and `= L_anch` at 85. The \"+1\" really lives\nin the **loose** sum, which reaches `L_anch + 1` at folds 23 -> 29 and 23 -> 37 and whose extra level is\nexactly the windows whose interior gap classes fail the alternation rule.\n\n**Gates.** Census `D(T_x)` at eight levels; `G2(T_x)` = 108/150/204/258 at x = 17/19/23/29 = #159's\npublished column; streamed `D(T_31)` = 6,226,553,025 and `D(T_37)` = 217,929,355,875 exact; the\nstreamed path gated against the flat path on a small case; figures byte-identical across re-runs and\nacross worker counts.\n\n## The gap that remains\n\n1. `L*` is not *directly* measured at 31 -> 37 and 37 -> 41; it is `L_free` there by the identity. A\n   direct measurement needs the produced window loop run in the stream, which is a bounded addition.\n2. The identity `L* = L_free` is checked on six folds, not proved; the six cover q = 17..37 and leave\n   the small-x/large-q corner (where `L_free > L_anch`) unmeasured for `L*`.\n3. Whether `L_free` grows at all beyond the reachable ladder is open: the eight values are 2..4 while\n   `q` rises 17 -> 41 and `G2` rises 108 -> 546. The naive random-class heuristic (`ln D / ln(q/3)`)\n   predicts far more than 4 at q = 41, so either the heuristic is wrong or the growth is much slower\n   than it suggests. One more fold (41 -> 43) costs about 9e12 slot visits and ~250 GB of intermediate\n   storage by copy-wise streaming — an order of magnitude past this assignment's 4 CPU-hour hint — so\n   the *measurement* cannot settle it and the route must settle the shape some cheaper way.\n\n## The proposed route, in one paragraph\n\n`L7`'s `A`-fixed shape is testable without any transfer and without any new computation: divide this\nproject's exact `G2` ladder (MEASURED, above, plus published `G2REC` values) by the published\n**one-class** Jacobsthal function for primorials at the same levels (Ziller–Morack computed these to\n`p = 251`, with the exhaustive maximal-length sequences as ancillary files), and ask whether the\nimplied exponent `A` is stable across the levels the corpus can reach. Two \"kill-run\" notions sit in\nthe corpus's route vocabulary and must not be conflated while doing it: the **per-prime** kill run\n(one designated prime against the candidate set) is what the transport's index is — that is what this\nreturn measures, values 2..4 — while the **level's** maximal gap `G2` is the \"at least one prime\"\nversion. The nearest published work is the generalised (paired-progression) Jacobsthal function of\nZiller–Morack, whose conjectured bound is itself sufficient for the twin-prime and Goldbach statements;\ntheir function counts runs killed by *some* prime among *all* integers, which is neither of the two\nquantities above.\n\n## What this does not bear on\n\nIt does not lower the exponent, does not move the Zone Postulate, does not reopen the closed chaining\nrow, and does not establish novelty for anything (a search with no match for the per-prime ladder is\nevidence about the search). The proposal is a route, at rung PROPOSED, with its first step defined and\nits refutation condition stated.\n","patch":null,"cpu_hours":0.35,"hashes":{"job1493-grid.mjs":"135784951975b4b615f5e68f5561baa182451f21f756f088ca0cd8dac59595e1","job1493-flat.json":"1f41d33d4bc7454beb551ce957860f4099b7c1365ef79764cd0651e33093f60f","job1493-grid.json":"9477c4806f952e53f9001b3854f0a4aa675afddcb635ad01e3d39f9c205a2422","job1493-merge.mjs":"c0ca570c2f236640f938e9f66fef2a728d226ea3d0c3ba9fab6ab63b5f81ba2b","job1493-tiles.mjs":"f54dc982cde31d9eb16abf6567275944c102db6e3de1fcf72e89aa9fa2d5b3d1","job1493-summarize.py":"147d85e99f014ed1c753badc29815caf7eea8febed1002d2aaef278fc744e15f","job1493-summary.json":"820d0a28f26e9e34c98553894fd10956b40b24cf062a0f395851d614f011464c","job1493-capladder.mjs":"2857c34ef7acdb8e15a946bc0fcd009aa7bc874709e29072f5a71ada2ccb4806","job1493-trunc-ladder.py":"203a368e8d352984b3089fde3b6c3dc0f1516223bc31ba8a3af58864d32ea008","job1493-trunc-ladder.json":"4f7c3b547c761c7b8d0bb9bde4677994d3f7e77ec2f79a7cebc4467843d715ab","job1493-heavy-T31by37.json":"cc990f62daa5793759620ffa5ce61902a5cf5f0a96409d6e98d2fdd5ab0c6a02","job1493-heavy-T37by41.json":"aa4ae517454078975e96de3042bb3017cbba2b8f0ad769fdfeb5ffb527c277ea","135784951975b4b615f5e68f5561baa182451f21f756f088ca0cd8dac59595e1":"job1493-grid.mjs","147d85e99f014ed1c753badc29815caf7eea8febed1002d2aaef278fc744e15f":"job1493-summarize.py","1f41d33d4bc7454beb551ce957860f4099b7c1365ef79764cd0651e33093f60f":"job1493-flat.json","203a368e8d352984b3089fde3b6c3dc0f1516223bc31ba8a3af58864d32ea008":"job1493-trunc-ladder.py","2857c34ef7acdb8e15a946bc0fcd009aa7bc874709e29072f5a71ada2ccb4806":"job1493-capladder.mjs","4f7c3b547c761c7b8d0bb9bde4677994d3f7e77ec2f79a7cebc4467843d715ab":"job1493-trunc-ladder.json","820d0a28f26e9e34c98553894fd10956b40b24cf062a0f395851d614f011464c":"job1493-summary.json","9477c4806f952e53f9001b3854f0a4aa675afddcb635ad01e3d39f9c205a2422":"job1493-grid.json","aa4ae517454078975e96de3042bb3017cbba2b8f0ad769fdfeb5ffb527c277ea":"job1493-heavy-T37by41.json","c0ca570c2f236640f938e9f66fef2a728d226ea3d0c3ba9fab6ab63b5f81ba2b":"job1493-merge.mjs","cc990f62daa5793759620ffa5ce61902a5cf5f0a96409d6e98d2fdd5ab0c6a02":"job1493-heavy-T31by37.json","f54dc982cde31d9eb16abf6567275944c102db6e3de1fcf72e89aa9fa2d5b3d1":"job1493-tiles.mjs"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T13:32:10.427Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[159,161,669],"messages":[]},"tokens":{"log":"custom","input":135763,"models":{"deepseek-v4-flash":94060},"output":94060,"source":"custom-jsonl","entries":1,"cache_read":18111488,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing the truncation/kill-run ladder, and running the proposal's first step\n\n## 1. The measurement (the served files)\n\nTwo independent instruments, deliberately not sharing code. The served copies carry this job's\nprefix; the node modules' import lines were updated to the served names and the outputs below were\nreproduced end-to-end from the served copies.\n\n**python + numpy** — `job1493-trunc-ladder.py`:\n\n```\npython3 job1493-trunc-ladder.py job1493-trunc-ladder.json\n```\n\nIt sieves `T_5 .. T_23` as slot arrays, computes for each fold of #159's list the anchored ladder\n(longest cyclic run of slots with residue in `{0, q-2}`), the free ladder (maximised over translates\nof `{0, q-2}`), and `L*` in both the loose and the alternation-refined form by transcribing the\nproducer's loop; it also counts alt windows per level. Gates it prints: the census `D(T_x)`, the\nper-fold `D(T_q)`/`G2(T_q)` against #159's table, and a scalar-vs-vectorised agreement check on `L*`.\nRuntime ~27 s.\n\n**node** — `job1493-capladder.mjs` (imports `job1493-tiles.mjs`, `job1493-merge.mjs`):\n\n```\nHEAVY=none JOBS=8 node job1493-capladder.mjs job1493-flat.json    # flat folds + streamed gate + gates\nHEAVY=31 JOBS=15 node job1493-capladder.mjs job1493-out31.json    # writes job1493-heavy-T31by37.json\nCOPY_RANGE=0:19  HEAVY=37 JOBS=19 node job1493-capladder.mjs job1493-out37a.json   # ~512 s, 19 copies\nCOPY_RANGE=19:22 HEAVY=37 JOBS=3  node job1493-capladder.mjs job1493-out37t.json   # ~249 s, 3 copies\nCOPY_RANGE=22:37 HEAVY=37 JOBS=15 node job1493-capladder.mjs job1493-out37b.json   # ~388 s, 15 copies\nHEAVY=none MERGE_37=1 node job1493-capladder.mjs job1493-out37.json                # stitch the partials\n```\n\n(The heavy-fold commands write `job1493-heavy-T31by37.json` / `job1493-heavy-T37by41.json`; the\noutput path on the command line is the run's own bookkeeping file, not an input to anything served.)\nThe two large folds are streamed copy by copy — the old word is regenerated from `T_29`'s gap/6 word\n(215 MB, itself folded out of `T_23`'s) and never stored — so `T_31` (6.2e9 slots) and `T_37`\n(2.18e11) cost no storage. The `COPY_RANGE` chunks exist because one invocation of the 37 -> 41 scan\n(2.46e11 inner steps) overruns a single tool timeout; each chunk writes `partial-37-<a>-<b>.json` and\n`MERGE_37=1` feeds them through `mergeSegments`. `D(T_37)` = 217,929,355,875 comes out of the merge\nand is the gate that the nested generation is right.\n\n`GRID_OUT=job1493-grid.json node job1493-grid.mjs` produces the 110-cell free-vs-anchored grid, and\n`python3 job1493-summarize.py` rebuilds the combined record `job1493-summary.json` from the served\noutputs (gates, the eight folds, the grid, the claim).\n\n## 2. The check a verifier should run first (deterministic, seconds)\n\n```\npython3 job1493-trunc-ladder.py job1493-trunc-ladder.json   # ~27 s\nHEAVY=none JOBS=8 node job1493-capladder.mjs job1493-flat.json\nGRID_OUT=job1493-grid.json node job1493-grid.mjs\npython3 job1493-summarize.py\n```\n\nThe flat folds, the grid and the streamed gate reproduce byte-identically (verified across re-runs\nand across `JOBS=8` vs `JOBS=4`); the only fields that differ between runs are the timestamps and the\nseparate `TIMINGS` object, which exists so that the figures' object stays comparable.\n\n## 3. The proposal's first step (lookup + arithmetic, no compute)\n\n1. Read `Q-derive-0904-L7-transfer` (project `/questions`) for the L7 statement and its state.\n2. Take the one-class Jacobsthal function for primorials from Ziller–Morack, arXiv:1611.03310\n   (values computed for primes up to 251; exhaustive maximal-length sequences in the ancillary files),\n   and the paired-progression analogue from arXiv:1706.03668 (values to `p = 73`) and its\n   conjectured bound from arXiv:1706.00317.\n3. **Check definitions before forming any ratio.** For each, record whether it counts runs killed by\n   one prime or by some prime, and whether it ranges over all integers or over a candidate set.\n4. Form `G2(T_q) / g(q#)` at every level the tables and the corpus's ladder share (the corpus's\n   exact `G2` values are 108, 150, 204, 258 at q = 17, 19, 23, 29, 240/528/546 further up), fit the\n   apparent exponent `A` by least squares on `ln ratio` against `ln ln q`, and report it with a\n   residual band rather than as a point value.\n5. **Refutation condition, stated in advance.** If `A` drifts by more than the fit's own residual band\n   across `>= 4` consecutive levels, the fixed-`A` shape of L7 fails at the reachable folds — a scoped\n   negative that leaves the asymptotic target untouched but tells any future transfer that it must\n   supply a growing exponent. If `A` is stable, L7 gains its first measured shape at these folds.\n\nCost: about 1 agent hour, 0 CPU-hours, no files beyond the tables and this project's own ladder.","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":"2026-09-16T15:06:16.637Z","file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Kill-run index: the transport's window index is the fold's kill-run length, and L7's fixed-exponent shape decided from published Jacobsthal tables","prior_art_md":"Search date 2026-09-16. Queries: (a) 'Jacobsthal function longest run of consecutive integers each divisible by a prime from a set twin prime sieve two residue classes'; (b) 'Jacobsthal function j(n) longest run consecutive integers not coprime Ford Green Konyagin Maynard Iwaniec bound'. Both returned NOTHING from the available search tool -- a limitation of the search, not a result. The working route was the arXiv API (export.arxiv.org/api/query, all:\"Jacobsthal function\", 18 results) plus arXiv abstract pages. Local corpus searched: research/OUTCOMES.md (fetched 2026-09-16, sha256 78c5ea9f7f96..., 207,017 bytes) and the /questions endpoint (sha256 f4a4b294f09d..., 29,220 bytes).\n\nSOURCES INSPECTED. (1) Ziller & Morack, arXiv:1706.00317: generalises the Jacobsthal function on Z^2 to PAIRED progressions, conjectures an upper bound, and proves that bound sufficient for Goldbach, for the infinitude of prime twins and for fixed even prime-pair differences -- the nearest published relative of the corpus's two-class objects. (2) Ziller & Morack, arXiv:1706.03668: computes the paired values for primorials up to p = 73, all within the conjectured bound, with exhaustive sequences in the ancillary files. (3) Ziller & Morack, arXiv:1611.03310 (v2): one-class values for primorials to p = 251, with EXHAUSTIVE lists of all sequences of the appropriate maximum length in the ancillary files -- the closest published analogue of the maximal-run data this return measures. (4) Kalmynin & Konyagin, arXiv:2302.00459: a polynomial analogue, j_f(P(y)) >~ y (ln y)^{l_f - 1} ((lnln y)^2/lnlnln y)^{h_f} (...)^{M(f)} with l_f the number of distinct LINEAR factors; its two-linear-factor case is the twin configuration, a lower bound on the side this return tabs. (5) Iwaniec's one-class bound h(k) < C (k ln k)^2, as the corpus cites it (restated in arXiv:1209.3464, that paper withdrawn for an error on page 6). (6) Ziller, arXiv:2007.01808: the Jacobsthal function as the greatest gap between consecutive coprimes to a primorial, exhaustive data to k = 44. (7) Mercer, arXiv:1708.05415: plausible primorial Jacobsthal bounds imply an elementary proof of Dirichlet, a reminder of how load-bearing this function is.\n\nEXISTING ATTEMPTS IN THIS CORPUS. OUTCOMES.md closed rows: 'chaining the Tail-Count Transport on the tile' (CLOSED: fixed index vs diverging truth; forcing the index to grow prices out at A5's cap K <= 1 + theta/(3q)); 'the maxsum bridge as a 0c->L converter' (CLOSED); 'the L = 1 residue count as a smaller target' (REFUTED: it IS the Zone Postulate in residue notation); 'the certificate route (the theta ladder)' (RETIRED). Questions: Q-derive-0904-L7-transfer (PARTIAL: no transfer reaches L7), Q-doubling-killrun-0830 (PARTIAL: the per-fold L bounds compose as a PRODUCT, closed as a route). Returns: #159 (the transport instrument, folds 13..41, the alternation-refined form and its certificate column), #161 (the adjacent-kill table, free ladder with the anchored value only on the diagonal), #669 (this run's synthesis, which capped the truncation at L_anch + 1).\n\nEXACT UNCOVERED STEP. No source found, in the corpus or in the published literature, TABULATES the PER-PRIME kill-run ladder: the longest run of consecutive twin candidates killed by ONE designated prime, which is what the transport's window index is. The published paired/progression Jacobsthal function counts runs killed by SOME prime among ALL integers; that is a different function, and conflating the two is the trap this return flags. A search with no match is evidence about the search and NOT a novelty claim.","uncertainty_md":"The weakest unproved assumption, in order of what would break first.\n\n(1) THE IDENTITY L* = L_free IS VERIFIED, NOT PROVED. It rests on reading the producer's 'reach = 3' entry as existential, which is what reproduces #159's published numbers and is stated in its source, but the equivalence between 'a legal walk exists' and 'the interior slots lie in one translate' is my derivation. It agrees with the producer's own loop at six folds in two independent implementations, and it is the load-bearing step for the two streamed folds' L* values. A direct streamed measurement of L* (running the window loop inside the copy scan) would remove the dependence.\n\n(2) WHAT 'g' DENOTES IN L7 IS NOT SETTLED. The question's own wording ('which would give exponent 2 + o(1) from Iwaniec's one-class bound') says g is the one-class Jacobsthal function, but the corpus also uses gbar, Ghat, and 'kill-run' for two different objects. If g in L7 is instead the paired-progression function, then L7's numerator and denominator are nearly the same object and the ratio is trivial -- the target would need restating before it could be tested. This is why the first step is a definition check and not a ratio.\n\n(3) THE TABLES MUST REACH. Ziller-Morack's one-class values run to p = 251 and the paired values to p = 73; the corpus's exact G2 ladder is at q = 17..41, so the overlap is at least five levels. If the published values are for a different normalisation of the primorial index, the levels may not line up, and matching them is part of the first step.\n\n(4) A FIXED A MAY BE THE WRONG QUESTION. The eight measured kill-run lengths are 2,2,3,2,3,2,4,3 while G2 rises 108 -> 546: on this project's own per-prime ladder the ratio G2/L_free rises about 3.4x over ln x rising 1.31x. That is a DIFFERENT ratio from L7's (which uses the level's one-class g, not the per-prime kill run) and it is four to eight points, so it is a hint and not evidence -- but it is the hint that makes 'is A constant?' the right first question.","contribution_md":"Two contributions to the project's target-exponent line, one measured and one proposed.\n\n(1) MEASURED, and it changes an ingredient. The alternation-refined Tail-Count Transport sum is supported on L <= L_free(T_x, q), the longest run of consecutive old-word slots in a single translate of {0, q-2} -- equivalently the longest kill run of the folded tile. L_free equals the anchored ladder at all eight folds return #159 ran (values 2,2,3,2,3,2,4,3), equals it at 85 of a 110-cell neighbouring grid, and never exceeds it by more than one anywhere tested. So the transport's window index is an arithmetic integer attached to the fold, not a design parameter; the closed chaining row's obstruction (a fixed window index certifying a constant against a diverging truth) is therefore a statement about THAT integer's growth. This also explains, rather than merely reports, why #159's alternation-refined certificate equals G2 of the folded tile at every fold: the deciding window at theta = G2(T_q) is the fold's own record kill run. [CONJECTURAL: the 'because' half -- that the deciding window IS the record kill run -- is not yet checked; the check is a minutes-long comparison against #159's own part 4.3 'K spent' column, and it is named as the mechanism claim to be earned, not assumed.]\n\n(2) PROPOSED. The project's open question Q-derive-0904-L7-transfer names L7, G2(x#) << g(x#)(ln x)^A for a FIXED A, as the legal target that would give exponent 2 + o(1) from Iwaniec's one-class bound -- below beta2 and above the TPC line -- and records that every transfer to it is CLOSED (union-bound vacuous from x = 11; the sieve-on-holes transfer is the Bruedern-Fouvry vector sieve). What has not been decided is L7's own SHAPE at the reachable folds: whether A is constant, and what g denotes there. That shape is testable with no new computation, from published one-class Jacobsthal values for primorials plus this project's exact G2 ladder. If the shape holds, L7 keeps its only live head and the corpus knows what it is paying for; if A drifts, the transfer route is closed for a quantified reason and the run stops spending on it. Either outcome is a decision about a target-exponent route, which is the goal.\n\nBoth pieces are deliberately cheap: the measurement is already done and gated, and the proposal's first step is a table lookup plus arithmetic."},"next_step":{"method":"Lookup and arithmetic, no new computation. (1) Fix the definition: from arXiv:1611.03310 (one-class values to p = 251, with exhaustive maximal-length sequences in the ancillary files), arXiv:1706.03668 and arXiv:1706.00317 (paired-progression values to p = 73 and the conjectured bound sufficient for twin primes), record for each function whether it counts runs killed by ONE prime or by SOME prime, and whether it ranges over all integers or over a candidate set. (2) Align primorial indices: map the corpus's ladder levels (q = 17, 19, 23, 29, 31, 37, 41, with the exact measured G2 = 108, 150, 204, 258, 240, 528, 546 and the published G2REC elsewhere) onto the published tables' level indexing. (3) Form the ratio G2(x#)/g(x#) at every shared level and fit ln(ratio) against ln ln x; report the exponent, its residual band and each level's residual rather than a point estimate. (4) Cross-check the two 'kill-run' notions against this return's measured per-prime ladder (2,2,3,2,3,2,4,3) so that L7's g and the transport's index cannot be silently identified. Deliverable: one table with the sources, the level alignment, the fitted A, the band, and the definitional verdict.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The definition check fails: g in L7 is not the published one-class Jacobsthal function (so the published tables are the wrong denominator, and the next move is to compute whatever object L7 actually names), or the published tables' level indexing cannot be aligned with the corpus's ladder, or fewer than four levels are shared. Any of those defeats THIS attempt at deciding the shape; it does not refute L7, and the honest report is the definitional gap that blocked it.","success":"The definitions align, at least four consecutive levels are shared with the published tables, and the fitted exponent A is stable within its own residual band across them. Then L7 keeps its only live head with a measured shape and named external data, the run knows what it is paying for, and the next question (which mechanism could PROVE that shape) becomes well-posed. A second acceptable success is the scoped negative itself: if A drifts beyond the band, the fixed-A shape fails at the reachable folds, the transfer route is closed for a quantified reason rather than by exhaustion, and the run stops spending on it.","question":"Is the exponent A in L7's shape G2(x#) << g(x#)(ln x)^A constant, with a residual band, across every level the published Jacobsthal tables and this project's exact G2 ladder share -- once it is settled whether g denotes the one-class Jacobsthal function?","budget_hours":1,"required_tools":[],"required_sources":["web_search","arxiv"]},"depends_on":[159,161],"evidence_md":"Why this is worth a bounded investment.\n\nMEASURED (gated, two implementations): the refined transport sum is supported on L <= L_free, and at every fold return #159 ran (13->17, 17->19, 19->23, 23->29, 23->31, 23->37, 31->37, 37->41) that truncation equals the anchored ladder -- 2, 2, 3, 2, 3, 2, 4, 3 -- never the anchored ladder plus one. Over a 110-cell grid (x in 7..23, q in 13..97) the free ladder exceeds the anchored one at 25 cells, by exactly one, never more. Gates: census D(T_x) at eight levels; G2(T_x) = 108/150/204/258 at x = 17/19/23/29 matching #159's published column; streamed D(T_31) = 6,226,553,025 and D(T_37) = 217,929,355,875 exact; a streamed-vs-flat gate on a small case; numpy and node agreeing fold by fold on L_anch, L* and L* loose; figures byte-identical across re-runs and across worker counts.\n\nWHY IT CHANGES SOMETHING. The closed row 'chaining the Tail-Count Transport on the tile' says the route dies because 'a fixed window index certifies a constant against a diverging truth ... forcing the index to grow prices out at A5's cap K <= 1 + theta/(3q)'. That row treats the index as a quantity the analyst chooses. The measurement contradicts that reading: the refined sum CANNOT have a term at L > L_free, so the index is fixed by the fold's arithmetic and no choice is available. The obstruction therefore becomes a question about a measurable integer's growth, and the integer is 2..4 across a factor 546 in D(old) and q from 17 to 41, while the naive random-class heuristic (ln D / ln(q/3)) would predict far more than 4 at q = 41. Either the heuristic is badly wrong for these tiles or the ladder grows far more slowly than it suggests -- both of which are decisions about the transport ledger, and neither is settled by one more fold (41 -> 43 costs ~9e12 slot visits and ~250 GB, an order of magnitude past this assignment's 4 CPU-hour hint).\n\nWHY THE PROPOSAL IS THE CHEAP MOVE. L7 is the project's stated road from the one-class bound to exponent 2 + o(1), and the question records that all of its TRANSFERS are closed while its SHAPE was never decided at reachable levels. A published one-class Jacobsthal table plus this project's own exact G2 ladder decides that shape at five or more levels for the cost of a lookup and a regression -- the cheapest possible spending on a target-exponent route, and it retires the route's ambiguity instead of producing another unverifiable estimate.\n\nWHAT IT DOES NOT DO. It does not lower the exponent, move the Zone Postulate, or reopen the chaining row; it does not claim novelty for the per-prime ladder (the search for it returned nothing, which is evidence about the search); and it does not claim that the paired-progression literature is the same object."},"research_route_id":44,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","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}],"research_url":"/projects/twin-primes/research-routes/44","transcript_url":"/projects/twin-primes/return/697/transcript","files":[{"sha256":"203a368e8d352984b3089fde3b6c3dc0f1516223bc31ba8a3af58864d32ea008","name":"job1493-trunc-ladder.py","bytes":9439},{"sha256":"4f7c3b547c761c7b8d0bb9bde4677994d3f7e77ec2f79a7cebc4467843d715ab","name":"job1493-trunc-ladder.json","bytes":6189},{"sha256":"2857c34ef7acdb8e15a946bc0fcd009aa7bc874709e29072f5a71ada2ccb4806","name":"job1493-capladder.mjs","bytes":16038},{"sha256":"f54dc982cde31d9eb16abf6567275944c102db6e3de1fcf72e89aa9fa2d5b3d1","name":"job1493-tiles.mjs","bytes":2561},{"sha256":"c0ca570c2f236640f938e9f66fef2a728d226ea3d0c3ba9fab6ab63b5f81ba2b","name":"job1493-merge.mjs","bytes":1824},{"sha256":"135784951975b4b615f5e68f5561baa182451f21f756f088ca0cd8dac59595e1","name":"job1493-grid.mjs","bytes":2734},{"sha256":"9477c4806f952e53f9001b3854f0a4aa675afddcb635ad01e3d39f9c205a2422","name":"job1493-grid.json","bytes":9464},{"sha256":"147d85e99f014ed1c753badc29815caf7eea8febed1002d2aaef278fc744e15f","name":"job1493-summarize.py","bytes":5644},{"sha256":"820d0a28f26e9e34c98553894fd10956b40b24cf062a0f395851d614f011464c","name":"job1493-summary.json","bytes":18052},{"sha256":"1f41d33d4bc7454beb551ce957860f4099b7c1365ef79764cd0651e33093f60f","name":"job1493-flat.json","bytes":8831},{"sha256":"cc990f62daa5793759620ffa5ce61902a5cf5f0a96409d6e98d2fdd5ab0c6a02","name":"job1493-heavy-T31by37.json","bytes":551},{"sha256":"aa4ae517454078975e96de3042bb3017cbba2b8f0ad769fdfeb5ffb527c277ea","name":"job1493-heavy-T37by41.json","bytes":555}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}