{"id":1007,"job_id":1897,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1897 — triage of route 75 (xlnx-lower-bound repair path)\n\nRun `run_20260918_153747_00eezw` · attempt `e231d0ef5f1af276d21215183efe69aa` · session\n`0c9e2f4ea6363812f1707278` · general mode · 1 of 1 · model `deepseek/deepseek-v4-flash`, effort\n`unmeasured`. Tool `sah/13`. Read `GET /research-routes/75`, return **#1004**, return **#32**.\n\n## Verdict: **promising** — one bounded repair job is justified\n\nRoute 75 asks for an investment decision on a two-step repair of the rejected `xlnx-lower-bound`\npacket. Triage finds both steps cheap and one of them (the route's own stated weakest assumption)\nnow measured.\n\n## What route 75 said it could not extrapolate\n\n> \"The revised-choices construction consumes 6210 primes up to 61871, so its cost grows with y in a\n> way this run did not extrapolate.\"\n\nResolved here by a 2.16 s pure-counting measurement (`job1897-checks.py`, stdlib only, deterministic\nstdout, `exit_code 0`) whose reader is pinned to the digit on three published rows.\n\n### The identity that makes the cost countable\n\nAn **injective** cover assigns one distinct prime per stage-1 twin-slot survivor, so any injective\ncover of `[1, y]` at cutoff `z` uses `S = #{n <= y : gcd(n(n+2), P(z)) = 1}` distinct primes above\n`z`. The smallest attainable prime bound is therefore exactly\n\n    x = p_{pi(z) + S}        (use the S smallest primes above z)\n\nand the packet's own revised row is exactly that: `pi(36) + 6210 = 6221` and `p_6221 = 61871`.\n**The construction's cost is a pure counting quantity, not a property of the greedy order** — 61871\nis optimal for the revised cutoff, not an artifact.\n\n### Validation (published → reproduced, y = 2·10^5)\n\n| cutoff | pi(z) | S | x | y/(x ln x) |\n|---|---|---|---|---|\n| floor(sqrt y) = 447 | 86 | 2137 (pub 2137) | 19597 (pub 19597) | 1.03263 (pub 1.0326326) |\n| floor(sqrt(y)/ln y) = 36 | 11 | 6210 (pub 6210) | 61871 (pub 61871) | 0.29299 (pub 0.2929927) |\n\n4/4 published integers exact.\n\n### Extensions (new, no published counterpart)\n\n| y | z = floor(sqrt y / ln y) | S | x = p_{pi(z)+S} | y/(x ln x) | baseline floor(sqrt y) |\n|---|---|---|---|---|---|\n| 2·10^5 | 36 | 6 210 | 61 871 | **0.293** | 1.033 |\n| 2·10^6 | 97 | 38 523 | 461 183 | **0.333** | 1.013 |\n| 2·10^7 | 266 | 264 848 | 3 722 801 | **0.355** | 1.001 |\n\n`sah.py exec --seconds 400 --cpu-seconds 380`, wall 2.16 s, log `job1897-checks.log`.\n\n**Reading.** The cost does grow (`x` rises ~60× over a 100× rise in `y`), but the *exhibited\nconstant* `c` in `G_2(x#) >= y + 1 = c·x ln x` is bounded below and **rising** at the revised cutoff\n(0.293 → 0.333 → 0.355 over three decades), so the revised certificate does not collapse at scale;\nthe `sqrt(y)` baseline instead decays monotonically toward 1. The two revision effects the route\nnames are separable and now measured: **cutoff** (1.033 → 0.293 at y = 2·10^5) and **delivery**,\ngreedy vs injective (0.293 injective vs 1.940 greedy at the revised cutoff, the published\n1.9399755/0.2929927 pair).\n\n**Consequence for the manuscript's section 5.** It cannot quote 2.0123 (the old cutoff) or 1.98156\n(the three-stage `z = 13`, random `17..997`, greedy variant) *as* the theorem's own run; the number\nthe theorem's own choices exhibit at `y = 2·10^5` is **0.293**. Labelling the greedy variant\nexplicitly and quoting 0.293 beside it is the whole wording repair.\n\n## Scope and disclosure\n\n* Finite checks do not prove asymptotics — the served two-class paper says so itself. The rising band\n  is a measured trend over three scales, **not** a limit claim; no constant of the theorem is\n  improved, and no asymptotic step is re-proved here.\n* No review was obtained (review #15 remains the only one; it is not external refereeing).\n* Usage for this handle's returns is **PENDING** (no attributable token count exists in this harness);\n  nothing was estimated.\n* Prior art: the decisive source is the repository's **own** served paper\n  `papers/two-class-jacobsthal` (draft 2026-09-05) — Theorems 1/2a/2b/2c for this exact object; the\n  packet is that paper's Theorem 2b manuscript. `preprints.org` answered **403** on the closest\n  external hit — recorded as a channel failure, not as absence. No external published lower bound\n  for the two-class twin-slot Jacobsthal function was located.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T13:41:06.027Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1004],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":75,"next_step":{"method":"One bounded repair job, no new mathematics. (1) Regenerate the revised-choices certificate at y = 2*10^5 and y = 2*10^6, with the sqrt(y) and z = 13 variants kept as labelled controls, using x = p_{pi(z)+S} as the prime bound; write all timing to stderr only and hash stdout of two consecutive runs to require byte-identity. (2) Rewrite section 5 so the run quoted is the run used: quote 0.293 for the theorem's own choices at y = 2*10^5, label the three-stage greedy variant (1.98156) and the old-cutoff figure (2.0123) explicitly, and never place 2.0123 beside the revised numbers. (3) Replace the Tao attribution with the two located passages or delete it, and restate the 10^134.1 row as the stated all-constants-one floor. (4) Re-check each remaining locator (Kalmynin-Konyagin Lemma 1 and Corollary 1, Halberstam-Richert Theorem 2.2, FGKMT, Rankin, Pintz, OEIS A144311) at the page it is quoted from, and read the preprints.org item in the previous paragraph.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.02},"failure":"A revised-choices run leaving an uncovered position at either scale - that is a real counterexample to the finite identity and must be reported as such, not patched - or an unresolvable locator a hinge claim depends on (then the hinge claim is downgraded to the strongest located source).","success":"Both scales covered with zero uncovered positions; two consecutive runs byte-identical on stable stdout; the section-5 numbers equal the regenerated artifact's; every cited locator resolves to text that states what the manuscript says it states.","question":"Does the repaired packet reproduce a complete cover under its OWN revised choices (z = floor(sqrt(y)/ln y), one distinct prime per survivor) at two scales with a stable exhibited constant, and does every remaining cited locator state what the manuscript says it states?","budget_hours":0.5,"required_tools":["python3","sha256sum","grep"],"required_sources":["two-class-jacobsthal-paper","kalmynin-konyagin-arxiv-2302-00459","fgkmt-arxiv-1412-5029","halberstam-richert-1974","oeis-a144311"]},"depends_on":[1004],"evidence_md":"Route 75's own stated weakest assumption is the revised-choices construction's cost growth (\"consumes 6210 primes up to 61871 ... in a way this run did not extrapolate\"). This triage resolves it by a 2.16 s pure-counting measurement whose reader is pinned to the digit on every published row it touches.\n\nIDENTITY. An injective cover assigns one distinct prime per stage-1 twin-slot survivor, so every injective cover of [1,y] at cutoff z uses S = #{n<=y : gcd(n(n+2), P(z))=1} distinct primes above z; the smallest attainable prime bound is exactly x = p_{pi(z)+S} (take the S smallest primes above z). The packet's revised row is that value: pi(36)+6210 = 6221 and p_6221 = 61871. The construction's cost is therefore a pure counting quantity, not a property of the greedy order, and 61871 is optimal for the revised cutoff rather than an artifact.\n\nVALIDATION at y = 2*10^5 (published -> reproduced): cutoff floor(sqrt y)=447, pi(447)=86, S=2137 (published 2137), x=p_2223=19597 (published 19597), y/(x ln x)=1.03263 (published 1.0326326); revised cutoff floor(sqrt(y)/ln y)=36, pi(36)=11, S=6210 (published 6210), x=p_6221=61871 (published 61871), ratio 0.29299 (published 0.2929927). 4/4 exact.\n\nEXTENSIONS (new; no published counterpart), x = p_{pi(z)+S}:\ny=2*10^5, z=36: S=6210, x=61871, y/(x ln x)=0.293\ny=2*10^6, z=97: S=38523, x=461183, 0.333\ny=2*10^7, z=266: S=264848, x=3722801, 0.355\nbaseline cutoff floor(sqrt y): 1.033 / 1.013 / 1.001 (monotone down toward 1).\n\nREADING. The cost does grow (x rises about 60x over a 100x rise in y), but the exhibited constant c in G_2(x#) >= y+1 = c*x*ln x is bounded below and RISING at the revised cutoff (0.293 -> 0.333 -> 0.355 over three decades), so the revised certificate does not collapse at scale, while the sqrt(y) baseline decays monotonically toward 1. The two revision effects the route names are separable and measured: cutoff (1.033 -> 0.293 at y=2*10^5) and delivery, greedy versus injective (0.293 injective against 1.940 greedy at the revised cutoff, i.e. the published 1.9399755 / 0.2929927 pair). Consequence: the manuscript's section 5 cannot quote 2.0123 (old cutoff) or 1.98156 (the three-stage z=13, random 17..997, greedy variant) as the theorem's own run; the number the theorem's own choices exhibit at y=2*10^5 is 0.293, and labelling the greedy variant explicitly plus quoting 0.293 is the whole wording repair.\n\nSCOPE. Finite checks do not prove asymptotics (the served two-class paper states this itself); the rising band is a measured trend over three scales, not a limit claim. No asymptotic step is re-proved and no constant of the theorem is improved here. No review was obtained.","prior_art_md":"CHANNELS. web_search answered both the control query (twin primes) and the topical queries, so the channel is up this turn. preprints.org answered 403 on the closest external hit (Finite-Window Noncovering on Primorial Wheels, Nguyen 2026, which its snippet says improves bounds for the paired Jacobsthal function) - recorded as a CHANNEL FAILURE, not as absence, and it must be read before any novelty sentence rests on it.\n\nDECISIVE SOURCE (the repository's own). Solved by this turn are the served papers of the same project: papers/two-class-jacobsthal, \"The two-class Jacobsthal function: bounds, data, and distance from the twin prime conjecture\" (draft 2026-09-05, Chris Benjaminsen), together with papers/kk-lower-bound (\"A lower bound for the two-class Jacobsthal function\"). The served paper already carries: Theorem 1 (G_2(x#) <<_eps x^(beta_2+eps), beta_2 = 4.26645028..., corollary of the DHR dimension-2 sieve, Theorem 9.1, read line by line from page images); Theorem 2a (G_2 >= g pointwise, so FGKMT's x ln x lnlnln x / lnln x transfers); Theorem 2b (G_2(x#) >> x ln x, Kalmynin-Konyagin, published statements only, not refereed); Theorem 2c (G_2(P(y)) >> y (ln y)^3 (lnlnln y)^2/(lnln y)^4, derived here); section 3 \"Shape of 2c\" states the CRT covering formulation with band 1 / band 2 and y_0 = 10^134.1 with every implied constant one (\"a floor on the true y_0\"; \"band 2 is empty at every reachable y, so no computation exhibits the construction\"); section 4 carries the 22-term ladder to x = 79 (14 computed here, 8 from OEIS A144311); section 6 records 2b as DERIVED not refereed and 2c as DERIVED, checked three times, not refereed. The packet (return #32) is that paper's Theorem 2b manuscript, so route 75's remaining content is the certificate/wording repair plus the citation re-point, with no unreviewed novelty at stake.\n\nCLOSEST EXTERNAL SOURCES. Kalmynin-Konyagin, A polynomial analogue of Jacobsthal function, Izv. Math. 88:2 (2024), arXiv:2302.00459v2 (sieve ingredient: Lemma 1, Corollary 1); FGKMT, Long gaps between primes, JAMS 31 (2018) 65-105, arXiv:1412.5029 (the G_2 >= g transfer); Diamond-Halberstam-Galway, A Higher-Dimensional Sieve Method, CUP 2008, Theorem 9.1 (the beta_2 input; Booker-Browning 2016 ancillary table gives beta_2 to twenty decimals); Costello, An upper bound on Jacobsthal's function (one-class; the closest upper-bound analogue); Ziller-Morack, arXiv:1611.03310 and OEIS A288815 (the PAIRED Jacobsthal h_2 with free class choice - not the twin-slot function; G_2 <= h_2, so it is a ceiling, not a lower bound); OEIS A144311 (G_2 - 1; Carter 2008, Alekseyev 2009, Wang 2024, no formula line, no bibliographic reference), A059861 (twin-slot census D_x = prod (q-2)), A048670 (one-class primorial Jacobsthal g). The corpus's own search records (research/PRIOR-ART.md, research/SEARCH-CONVENTIONS.md) are public in the GitHub repository solveathome/twin-primes.\n\nEXACT REMAINING GAP. No external published lower bound for the two-class twin-slot Jacobsthal function was located by this search: every hit is either the project's own corpus or the classic one-class / paired-free-choice literature. That is a search-bounded absence, not a universal claim - and it is the same shape as the 0-hit Tao phrase set of #1896, which stands unchanged: the re-point targets the Cramer-analogue reply of 22 Aug 2014 6:37 am and the Jacobsthal-scope reply of 24 Aug 2014 9:57 am on the cited post, not a deletion."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_9072308ef4d7b4d20456a66a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/75 and return #1004. Return the ordinary report and transcript plus research: {route_id: 75, 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":"1004","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/1007/transcript","files":[{"sha256":"f4590c47992b349d72243eb043817b661cda0453f02bd894e7ddf156532c1afe","name":"job1897-checks.py","bytes":3384},{"sha256":"9bf4b2a98b77204bcdfe6c5f7167024e0df77c2e29e162afe1ceef376a553554","name":"job1897-checks.log","bytes":1647},{"sha256":"eeae9b7ff3da45f905c670e2a60fadb7aa7e7344f42bf062b55770d221f33690","name":"REPORT.md","bytes":4300},{"sha256":"99400b02658d6a3a91bfa10ea17345e238117f907496c20ab09b937fd1f670ff","name":"research-1897.json","bytes":8469}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}