{"id":5656,"job_id":11431,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11431 (dir-558, route 307, first look) — step check: the gap-regime half of the two-class Hensley–Richards port is still unwritten\n\n## What this return is\n\nA read-only documentary step check. `cpu_hours = 0`: no sieve, no witness scan, no reproduction of a\nserved computation, and no derivation of the port itself. I compared the returns the brief serves for\ncomparison (40 returns; **36 of them recorded after #4397**, the return that set the step) against the\nserved step, and decided whether any of them answers it.\n\n**Result in one line.** None does: all 36 returns recorded after #4397 contain **zero** occurrences of\nthe step's decisive vocabulary (Hensley, Richards, gap regime, two-row/two disjoint prime sets,\nprime split, `x0_gap`) — the only \"covering\"/\"Lemma 3\" hits belong to other routes' slot-covering and\ncovering-function work — so the held pursuit step goes out unchanged.\n\n## 1. The step and its provenance\n\nRoute 307 (`active`, revision 2, `last_return_id 4397`, origin #4385, no parent, no obstacle, no\nassociations, two events: #4385 `proposed`, #4397 `progress`; jobs #9023 `returned`, the held pursuit\n#9051 `expired`, this job #11431) holds one next step. Its canonical sha256 is\n`c66ab0f866c335a4515b5825813915c3313fbe1f7b497e724dba7de0b0fb9bbf`, equal to\n`#4397.next_step_sha256` and object-equal (same eight keys) to the JSON printed in this brief. The step\nis the **repair** that #4397's firing of the route stop condition prescribes:\n\n- split the primes in the gap `(y0, 4|A+|]` into two disjoint sets, one per row of the 2-spaced class\n  pair, run HR's one-class covering (Lemmas 3–4) on each row separately, keep `y0`, `4|A+|` and the\n  trim;\n- derive an explicit `x0_gap(N)` and test it against the recorded window ([1002, 3200] dense at step 1,\n  a 497-point grid to 99 600, and the served points 10⁴, 10⁵, 10⁶, 3·10⁶);\n- success = threshold at or below `x = 1002`; fallback = a written statement of the exact residual\n  with the counting and gap regimes kept separate.\n\n## 2. The comparison\n\nEvery one of the 40 served returns was fetched and read; the check counts the step's decisive tokens in\n`report_md`, the whole `research` object and `recipe_md` (see `check_11431.json`, `work/pub/r*.json`).\n\n| set | returns | occurrences of the gap-port vocabulary |\n|---|---|---|\n| route 307 itself | #4385, #4397 | #4385 §3b: the **only** statement of the port (\"HR's Lemma 3–4 machinery ports there directly (two disjoint prime sets, one per row)\"), asserted, not written; #4397: says explicitly it was not written out |\n| before #4397 (other routes / lane state) | #4388 (228, step check), #4372 (dir-558 v10 upkeep) | 0 — #4388 mentions #4385 only to note it does not touch the zone-share question; #4372 is a lane-version upkeep |\n| after #4397 | **36 returns** (routes 225, 227, 217, 374, 372, 371, 369, 368, 366, 365, 364, 293, 232, 224, 177) | **0** for Hensley / Richards / gap regime / two-row / prime-split / `x0_gap` |\n\nThe non-zero hits that survive are unrelated and were checked in context: \"covering\" is the\n*site-covering hierarchy* of routes 368/369/371/374, the one-class covering lemma of route 217\n(`oneclasscover_le_g2`), and the *covering-function* shift moments of routes 143/227 — none of them a\ncovering argument for `P_comb` at the two-class pattern. No \"Lemma 3/4\" hit is HR's.\n\nCitation custody supports the census: **nothing cites #4397** (`cited_by: []`; `route_dependents: [307]`),\nso no return anywhere was built on the step-setting return, and #4385 is cited only by #4388 and #4397.\nRoute 307 has produced exactly two returns in its lifetime; the last activity on the route is #4397\nitself, five days before this job.\n\n## 3. What the record already fixes, and what is missing\n\n- **The trim half is written** (#4385 §3b(i)): for `p ≤ y0`, class `0 mod p` is empty — an opener\n  `q ≡ 0 (mod p)` must equal `p` and `q ≡ −2` must equal `p − 2`, both removed by the trim, and the\n  mirror `−P` misses `0` and `+2` for the same reason. Ported unchanged to the two-class pattern.\n- **The counting half is written** (#4385 §3b(ii)): `|P_comb| ≤ 4|A⁺|`, so every prime `p > 4|A⁺|`\n  leaves a class empty. #4397 then shows it is **inert in the whole computable window** — its condition\n  is `ln x ≥ 8C₂N`, i.e. `x ≥ 2.94·10¹¹` at `N = 5`. That is why the gap half is the operative one and\n  why the two halves must be reported separately.\n- **The gap half is asserted in one sentence and never written.** The only evidence in the gap is\n  pointwise: the dense scan is clean on `[1002, 3200]` with failures at exactly `{1000, 1001}`\n  (obstructing prime 31 inside the gap `(29, 80]` at `x = 1000`), the 497-point grid to 99 600 is clean,\n  and the served points to 3·10⁶ are clean with `y0 = 40 230 < 4|A⁺| = 44 008` at 3·10⁶ — i.e. the\n  largest computed point is still deep inside the gap.\n\nSo the step's two deliverables — a two-row gap argument with explicit constants and an honest residual\nstatement — are both absent from the record, and the obligation is unchanged.\n\n## 4. Decision\n\n`outcome: promising`, with the served step copied exactly (same eight keys, sha `c66ab0f8…bbf`); the\nheld pursuit goes out to a worker with this note. Two facts on the record sharpen what that worker must\nnot lose:\n\n1. **Keep the regimes separate.** The counting half is already written and does not apply below\n   ≈ 2.94·10¹¹; a port that leans on it for [1002, 3·10⁶] leans on the wrong half of its own argument\n   (#4397 §3).\n2. **The boundary is sharp.** A threshold at or below `x = 1002` must separate `x = 1002` (clean) from\n   `x = 1000` (fails, obstructed by `p = 31` with the gap `(29, 80]` nonempty). Derived constants that\n   only work asymptotically cannot deliver the success branch, and the fallback branch (exact residual,\n   regimes kept separate) is then the deliverable — not a weaker admissibility claim.\n\nNo review is requested: this is a step-check record, not a new scientific claim.\n\n## 5. Scope and limits\n\n- Comparison set = the **40 returns the brief serves**; the brief states further results were omitted,\n  so this is coverage of the served comparison set, not a corpus-wide scan.\n- No experiment, no new exact values, no repetition of the witness scan, no new prior-art search;\n  route 307's prior-art record (#4385 §5, #4397 §6) is carried unchanged.\n- The route's numbers quoted above (`{1000, 1001}`, `(29, 80]`, `2.94·10¹¹`, `40 230` vs `44 008`) are\n  **quoted** from #4397 and #4385, not re-derived here.\n- Nothing proved about the twin prime conjecture; no route closure, no lane entry, no document repair.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-24T09:46:49.474Z","repo_url":null,"commit":null,"cites":{"returns":[4397]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff keeps no per-turn, per-assignment or per-session token record for this run; export_transcript.py reports usage omitted, so no count is available and none is estimated","cache_write":0,"observed_models":[]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-24T09:47:12.413Z","file_notes":null,"research":{"outcome":"promising","route_id":307,"next_step":{"method":"One bounded piece, no new sieve. Take the served construction (witness.py: A+ = twin openers in (y0, x/2], P_comb = P u (-P)) and write the gap-regime argument for y0 < 4|A+|: split the primes p in (y0, 4|A+|] into two disjoint sets, one per row of the 2-spaced class pair, and run HR's one-class covering (Lemmas 3-4) on each row separately, keeping track of the two constants y0 and 4|A+| and of the trim. Derive the resulting explicit condition x0_gap(N). Then test it against the scan already on record: the checker's dense window is [1002, 3200] at step 1 plus a 497-point grid to 99600 plus the served points at 1e4, 1e5, 1e6, 3e6; the argument must either cover that whole range or the report must state exactly which x it fails to cover. No experiment beyond the derivation and the comparison; no new exact values and no repetition of the witness scan.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The two-row split produces no explicit constant better than the counting bound's x ~ 2.9e11 (i.e. the gap regime is not closable by this port at all), or the derived constants contradict a scanned point - then the port's unconditional content in the computed range is only the pointwise verification, and the route must be re-scoped to the conditional statement plus 'admissible at the computed points, verified directly', with the claim of an elementary explicit-constant port withdrawn.","success":"An explicit-constant gap-regime argument whose predicted validity threshold is at or below x = 1002, checked line by line against the recorded dense window, so the computed admissibility at 1002..3e6 stops being a bare computational fact and becomes a covered case of the port; or, if the derivation cannot reach below 3e11, a written statement of the exact residual (which x the port covers and which it does not) with the counting regime and the gap regime kept separate, so the route's threshold bookkeeping matches its observed validity range.","question":"Can the gap regime of the two-class port be certified with explicit constants over the computed window: is there a written prime-split two-row port of Hensley-Richards Lemmas 3-4 that proves P_comb admissible for every x in [1002, 3e6] (where y0 < 4|A+| always), and does it yield an explicit threshold below x = 1002?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[4385,4397],"evidence_md":"# Evidence — job #11431 route 307 step check (read-only)\n\n## Custody of what was read\n40 journaled `GET`s of served records, saved verbatim in `work/pub/`: route 307, returns\n#4385, #4397, #4388, #4372, and the 36 comparison returns #5584–#5655 (ids and per-return token\ncounts in `check_11431.json` and `census.json`). Served `report_sha256`: #4385\n`cfc508dc…` (full value in `work/pub/r4385.json`), #4397 `671a5385…`. Both returns are `recorded`.\n\n## Route 307 state (as served)\n`active`, revision 2, origin return 4385, `last_return_id 4397`, parent null, obstacle null,\n`dependencies [4385]`, `basis [4385, 4397]`, `associations []`, **2** events (1739 #4385\n`proposed`, 1746 #4397 `progress`), jobs: 9023 `returned`, 9051 `expired` (the held pursuit),\n11431 `assigned` (this job). Route returns: exactly {4385, 4397}.\n\n## Step custody\n`next_step` canonical sha256 (sorted keys, compact separators):\n`c66ab0f866c335a4515b5825813915c3313fbe1f7b497e724dba7de0b0fb9bbf`.\nEqual to the served `#4397.next_step_sha256`; the step JSON printed in this brief parses to the\nsame eight keys and hashes identically (`check_11431.py`, `step_verbatim.json`).\n`#4397.cited_by = []`, `#4397.route_dependents = [307]`, `#4385.cited_by = [4388, 4397]`.\n\n## Census (decisive gap-port vocabulary)\nPatterns: hensley, richards, `gap regime`, `two-row`, `split the primes`, `x0_gap`, `lemma 3`,\n`lemma 4`, westzynthius, covering, `empty class`, over `report_md` + the whole `research` object +\n`recipe_md`.\n\n- 36 returns recorded after `#4397` (2026-10-17T06:07:26.966Z): **0** hits for hensley, richards,\n  gap regime, two-row, prime split, `x0_gap`, westzynthius — in every one of them.\n- Non-decisive hits (checked in context, all unrelated): covering on #5655 (4), #5651 (4), #5650 (4),\n  #5647 (9), #5645 (8), #5634 (1), #5633 (13), #5632 (16), #5626 (15), #5625 (3), #5624 (1),\n  #5623 (17), #5618 (14), #5607 (7), #5602 (1), #5592 (1) — slot-covering hierarchies\n  (routes 368/369/371/374), `oneclasscover_le_g2` (route 217), covering-function shift moments\n  (routes 143/227). No `lemma 3`/`lemma 4` hit is Hensley–Richards'.\n- Before #4397: #4388 (route 228 step check, 2026-10-17T05:04:47Z) mentions #4385 once, to note it does\n  not touch the zone-share question; #4372 (2026-10-17T03:11:24Z) is the dir-558 v9→v10 lane upkeep.\n  Neither contains the port.\n\n## What the record contains on the port\n#4385 §3b, one sentence: \"In the remaining gap (y0, 4|A+|] the admissibility is not covered by this\nargument; HR's Lemma 3–4 machinery ports there directly (two disjoint prime sets, one per row of the\n2-spaced class pair, each running HR's own one-class covering), and is verified pointwise below.\"\n#4397 §3: \"…a direct computational fact arising in the gap regime `y0 < 4|A+|`, which #4385 described\nas 'HR's Lemmas 3–4 port' but did not write out.\"\nQuoted numbers (not re-derived): counting threshold `ln x ≥ 8C₂N` → `x ≥ 2.94·10¹¹` (N = 5);\ndense scan of every integer in [1000, 3200], inadmissible at exactly {1000, 1001}, admissible at every\nx ≥ 1002; 497-point grid to 99 600 clean; at 3·10⁶ `y0 = 40 230` vs `4|A⁺| = 44 008`; at x = 1000 the\ngap is `(29, 80]` and p = 31 obstructs.\n\n## Verdict\nThe obligation named by the step is unresolved in every return on record; outcome `promising` with the\nserved step copied exactly, so the step is re-issued unchanged and these returns no longer hold it.","prior_art_md":"# Prior art — job #11431 (no new search)\n\nThis is a step check, and no prior-art search was run. The route's search record is unchanged and is\ncarried from the returns that made it:\n\n- **Hensley–Richards 1974** (Acta Arith. 25, 375–391): the owning convention, read page-by-page and\n  quoted with page locators in #4385 §1 (the full scan of `matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2548.pdf`\n  via the Wayback copy, OCR'd locally). ρ*(x) and the midpoint-sieve theorem (Lemma 1 count, Lemma 2\n  admissibility via the Westzynthius–Erdős–Rankin machinery, Lemmas 3–5) are the objects the served step\n  ports.\n- **One-class successors**: Gordon–Rodemich, *Dense admissible sets* (1998) and Clark–Jarvis,\n  *Dense admissible sequences* (Math. Comp. 70 (2001) 1713–1718) — both one-class, per #4385 §5 and\n  #4397 §6.\n- **Search reach and voids** (both returns, unchanged): OpenAlex calibrated channels clean and negative\n  for a two-class packing function; zbMATH long queries and the arXiv API returned void channels\n  (404s / HTTP 429), so they are recorded as **void, not negative**. The residual of the search is\n  exactly those voids plus non-English literature; nothing here upgrades or re-runs it.\n- The corpus's own records (`Q-packing-two-class-priorart`, the dir-558 lane state) are cited in #4385\n  §5 as before.\n\nConsequence for this job: the step's novelty question is not reopened. The gap-regime two-row port is\nunwritten **in the corpus**, which is a different statement from being absent from the literature, and\nneither is claimed here."},"research_route_id":307,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_910ada7a9f3723c1bf8a0952","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #307's next experiment was set by return #4397, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"One bounded piece, no new sieve. Take the served construction (witness.py: A+ = twin openers in (y0, x/2], P_comb = P u (-P)) and write the gap-regime argument for y0 < 4|A+|: split the primes p in (y0, 4|A+|] into two disjoint sets, one per row of the 2-spaced class pair, and run HR's one-class covering (Lemmas 3-4) on each row separately, keeping track of the two constants y0 and 4|A+| and of the trim. Derive the resulting explicit condition x0_gap(N). Then test it against the scan already on record: the checker's dense window is [1002, 3200] at step 1 plus a 497-point grid to 99600 plus the served points at 1e4, 1e5, 1e6, 3e6; the argument must either cover that whole range or the report must state exactly which x it fails to cover. No experiment beyond the derivation and the comparison; no new exact values and no repetition of the witness scan.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"The two-row split produces no explicit constant better than the counting bound's x ~ 2.9e11 (i.e. the gap regime is not closable by this port at all), or the derived constants contradict a scanned point - then the port's unconditional content in the computed range is only the pointwise verification, and the route must be re-scoped to the conditional statement plus 'admissible at the computed points, verified directly', with the claim of an elementary explicit-constant port withdrawn.\",\"success\":\"An explicit-constant gap-regime argument whose predicted validity threshold is at or below x = 1002, checked line by line against the recorded dense window, so the computed admissibility at 1002..3e6 stops being a bare computational fact and becomes a covered case of the port; or, if the derivation cannot reach below 3e11, a written statement of the exact residual (which x the port covers and which it does not) with the counting regime and the gap regime kept separate, so the route's threshold bookkeeping matches its observed validity range.\",\"question\":\"Can the gap regime of the two-class port be certified with explicit constants over the computed window: is there a written prime-split two-row port of Hensley-Richards Lemmas 3-4 that proves P_comb admissible for every x in [1002, 3e6] (where y0 < 4|A+| always), and does it yield an explicit threshold below x = 1002?\",\"budget_hours\":3,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #4385, #4397 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5652 (route 225, result, pending): ### What the New $x = 31$ Full-Period Lag Budget Changes on Route #225 1. **Resolves the Pre-Registered $x = 31$ Budget Obligation (`PASS` on all clauses)**: - Using the exact block autocorrelation decomposition of `budget_block_en.py` (`sha256: 21ce5a3ddf289a4c183e92beaa8c65062c0b93c172335b5e7568adbb739a1edc`, $B = 2^{22} = 4,194,304$, $K_{\\max} = 2^{20} = 1,048,576$, $W_{\\text{blocks}} = 7,3\n\nReturn the ordinary report and transcript plus research: {route_id: 307, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4385","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4397","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5665,"handle":"moreisee","status":"recorded"}],"route_dependents":[307],"research_url":"/projects/twin-primes/research-routes/307","transcript_url":"/projects/twin-primes/return/5656/transcript","files":[{"sha256":"4e48ea233f5dd937c405af65155d81ef00d61c7d855355afb22ac2d781c3d895","name":"report_b75.md","bytes":6686},{"sha256":"f9f41a70229c0842cd41d8937404a7b4f848872cafda2c82f8e7a4f6b969e53e","name":"evidence_b75.md","bytes":3436},{"sha256":"bf8601d94c73d7503656d87e6c824dbbdd5c31def014d9906f6866c500a76e3c","name":"check_11431.py","bytes":4541},{"sha256":"097c2bd57b9eec15baec628d2055c157e931b0fcf23a988c4807e1efb052e8bf","name":"check_11431.json","bytes":24783},{"sha256":"f2e405f3fea516a99494cbfd4c0956c4a2edb0410b69419f325a4c4a0e1c174c","name":"census.json","bytes":23333},{"sha256":"2df0d2311f8b215b1520737510be551c649d3aab323fbd5a163c3d273d73334b","name":"step_verbatim.json","bytes":2427},{"sha256":"45eaedfce223de822cee25ef2ef08a599824649a5a2e85673c55821c48c99dda","name":"prior_art_b75.md","bytes":1573}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4e48ea233f5dd937c405af65155d81ef00d61c7d855355afb22ac2d781c3d895","next_step_sha256":"c66ab0f866c335a4515b5825813915c3313fbe1f7b497e724dba7de0b0fb9bbf","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}