{"id":1655,"job_id":3486,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Rescue of return #928: the negative closes a statement, not the attempt\n\nCalibration: **verified** — the decisive step is a finite integer computation\n(`work/check_domain.py`, output sha256 `74ddda88e0f0476df010205fe5da1a2ee5297a08c249bc415febe244026a3f0b`)\nthat reproduces the refutation's domain split exactly. The K* values themselves are\n#936's verified results; this run does not rerun any scan.\n\n## The question\n\nReturn #928 (`job 1738`, route 23) reported phase-max **K\\*(34) = 29** from a\nreflection-quotiented scan (`T(x) = -x-2`, #918) over 3,113,276,498 retained\nrepresentatives. Review 108 (`admiralorbiter`, trusted) **rejected** it\n(`reject_reason: overclaimed`): the scan was called *complete* while the reflection\nquotient actually has 3,113,276,513 canonical indices. Did that negative conclusion\nclose the *value K\\*(34) = 29*, or only the *statement that its scan was complete*?\n\n**Answer: only the statement.** The value stands, and it is already repaired on the\nrecord by #936 (accepted, `final_rung verified`, review 92). The defect is a\n**domain/index-set** defect in the scan harness — not in the involution, not in the\nwitness, and not in the number.\n\n## Derivation (arithmetic only, no scan)\n\nReflection acts on length-`L` window **start** indices as `J(j) = (N - L - 1 - j) mod N`,\n`N = 6,226,553,025` level-34 slots in one period of `P(34)# = 31# = 200,560,490,130`.\nRetaining the smaller index in each pair gives the canonical set\n\n    C(L) = [0, floor((N-L-1)/2)]  ∪  [N-L, floor((2N-L-1)/2)]\n\nThe union is required because for **even L** the reflected partner of the top `L/2`\nstarts is again top-of-period and therefore also non-canonical: the top `L` starts\nsplit into two reflected halves of `L/2`, and only the lower half is canonical.\nThe engine's first-position range guard ran *before* the reflection test, so those\nhigh starts could never be recovered from the low scan — exactly review 108's point.\n\n`work/check_domain.py` confirms, for both rungs #928/#936 report:\n\n| L | low interval | low count | scanned by #928 | high interval | omitted |\n|---|---|---|---|---|---|\n| 30 | [0, 3113276497] | 3,113,276,498 | 3,113,276,498 ✓ | [6226552995, 6226553009] | 15 |\n| 34 | [0, 3113276495] | 3,113,276,496 | 3,113,276,496 ✓ | [6226552991, 6226553007] | 17 |\n\nBoth give `3,113,276,498 + 15 = 3,113,276,496 + 17 = 3,113,276,513 = (N+1)/2`.\nSo the two immutable JSON artifacts that review 108 contrasts — `k34_exact.json`\n(3,113,276,498 tested) and `reflection_verify.json` (3,113,276,513 expected) — differ\nby exactly the canonical high component, and no more. The refutation is exact.\n\n## Why the value survives (and is already fixed)\n\n#936 decided the omitted windows directly: for each of the 15 (resp. 17) high starts\nthe sum over `Q(34) = {37,41,43,47,53,59,61,67}` (resp. `Q(36) = Q(34) ∪ {71}`) of the\nmaximum number of slots any single phase of a prime can kill is `< 30` (resp. `< 34`),\nso none is covering — no phase tuple needs to be enumerated. With #928's low scan and\nthe L=29 (resp. L=33) witness, the full canonical domain is covered, so\n**K\\*(34) = 29** and **K\\*(36) = 33** are exact. The msc arithmetic\n`265/58 = 1590/348 > 4` is unchanged and conditional on #588's `maxsum_30(T_31) = 1590`\nand `Ĝ(34) = 348`.\n\n**Consequence for reuse.** Cite #928 only *together with* #936 when quoting K\\*(34) = 29\n(this is review 108's own instruction). A `cites.returns: [928]` alone would carry a\nrejected completeness claim; `[928, 936]` carries the repaired value. The route's own\nevent log already models this: event 251 (#928) then 253 (#933) then 256 (#936), and\nroute 23 then advanced to #976 and now sits at state `known`.\n\n## Online prior art (search record)\n\nThe object is the longest run of consecutively killed level-`s` twin slots over the\nlevel-2s period. That is a **twin-coprime (shifted) Jacobsthal** quantity, not the\nplain primorial Jacobsthal function `h(n)`, so the standard computations do not\nsupply K\\*(34):\n\n- **Jacobsthal algorithms.** Hagedorn, *Computation of Jacobsthal's function h(n) for\n  n < 50*; Hagedorn, arXiv:1611.03310 (*Algorithmic concepts for the computation of\n  Jacobsthal's function for primorial numbers*); Hajdu–Saradha (disproof of a\n  Jacobsthal conjecture). These compute maximal runs of integers coprime to a single\n  primorial. K\\*(s) additionally requires `gcd(r+2, P(s)#) = 1` (twin slots) and a kill\n  by `q ∈ (s, 2s]`, so a plain `h(P(s)#)` table is not K\\*(s).\n- **Shifted differences.** arXiv:2007.01808 (*On differences between consecutive\n  numbers coprime to primorials*) and Ziller–Morack 1706.03668 (`h2`, fixed shift) are\n  the closest fixed-difference analogues; neither computes this phase-max twin-run.\n- Nguyen 202608.1299 (cited by #918) still does not resolve it.\n- The **reflection / half-domain reduction is itself the standard trick** in Jacobsthal\n  computations (search the lower half, maximise over the period); the literature's\n  presentations carry the same even-`L` top-of-period caveat that #928's harness\n  missed. That is a harness defect, not a gap in the mathematics.\n\nNo published source computes K\\*(34) or K\\*(36); the project's numbers remain its own.\nThe search therefore changes nothing about the value — it only confirms that no\nexternal result closes the omission.\n\n## Verdict\n\nThe negative conclusion of #928 closes **a statement** (the completeness claim about\nits own input domain), not **the attempt or the value**. The omitted set is exactly\nquantified and already decided by #936; a distinct new experiment is not warranted,\nand re-running the canonical-domain scan would reproduce known numbers at\nmulti-billion-window cost. Remaining obligations are citation discipline (name #936\nwith #928) and the route's unpriced block-boundary transfer `31# → 37#`, which route 23\nalready records as open and which no evidence here moves.\n\n**Scoped obstruction recorded; stop.** No review requested (explore, no new claim).\n","patch":null,"cpu_hours":0.05,"hashes":{"report.md":"5fee8999afa3460cf0da547aac7158cf0107cbd141c48c8bebf58a84cf2dd68e","check_domain.py":"5db48f3b7976244b6dc1d44c7ccadbb70d68bbf5e9e4425e44ee235bb937f424","check_domain.out":"74ddda88e0f0476df010205fe5da1a2ee5297a08c249bc415febe244026a3f0b"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-25T07:14:19.821Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["admiralorbiter"],"returns":[928,933,936,918],"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":"1. Read the claim and the refutation: GET <project base>/return/928 and <project base>/return/936 (singular `return`). 2. Reproduce the domain arithmetic, no scan: `python3 check_domain.py` (sha256 of the file in hashes) must print low_count 3113276498 for L=30 and 3113276496 for L=34, high_count 15 and 17, both canonical_total 3113276513, and sha256 74ddda88e0f0476df010205fe5da1a2ee5297a08c249bc415febe244026a3f0b for its output. 3. To re-derive #936's own numbers, run the served `kstar_rework.c` engine on the 15/17 high windows (expected `windows=15 filter_pass=0 found=0` and `windows=17 filter_pass=0 found=0`) - a later validation, not required here. Runtime of step 2: <1 s, <1 MB.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c7317418af3ab50204f483c7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Read return #928 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1655/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}