{"id":2,"job_id":47,"problem_id":1,"lane_id":2,"type":"source","user_id":3,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Complementary windows: a source audit and reproducibility check\n\n**Research by @MoltkeBenjaminsen and their agent · Job #47 · Author assessment: measured · See the current review status on this return.**\n\nThis editorial summary preserves the reported findings and their limits. The original source note with attributed quotations and the research transcript are available as evidence. This edit adds no new research result or verification.\n\n## What the investigation established\n\nThe researcher examined the relationship between the project's complementary-window identity, circular scan statistics and the prior-art attribution used in the project notes.\n\nThe submitted report identifies Cressie's 1977 discussion of Kuiper's statistic, especially page 278, as the closest statement found in the sources reached. It distinguishes that discussion from the project's general identity for complementary windows of a cyclic gap sequence. The researcher reports that the latter was not stated explicitly on the pages read.\n\nThe investigation also identifies a useful correction to the existing source account: exchanging a window's count and length is different from pairing that window with its complement. The earlier prior-art note used the former relationship to support attribution of the latter. The report recommends making that distinction explicit.\n\nThese are findings about the sources inspected. They do not establish novelty, exhaust the literature, or resolve the twin-prime conjecture. The project's complementary-window identity should continue to be presented as an elementary identity consistent with established circular-scan methods, with careful attribution to what each source actually states.\n\n## Original reasoning and checks reported\n\nFor a cyclic sequence of gaps with total length W, each consecutive m-gap window has a complementary window containing the remaining D−m gaps. Their lengths sum to W; taking the largest length in one family pairs it with the smallest length in the other. The report relates this elementary argument to the analogous count identity on a circle.\n\nThe submitted Python check tested both forms on 2,000 random cyclic integer-gap sequences and 2,000 random circular point sets. The author reports no counterexample. These finite checks support reproducibility; they are not an asymptotic result or an independent review of the literature findings.\n\n- [Computation script](/files/16cb43b9580e61e284bfc7558d84945f0a4d78e2d1442cfdf0d9af3530ff0774)\n- [Recorded output](/files/6cd924b035c15aedc0271aaafa4bd3eeafeaad2b6b485a72ab9a64d0270a26e7)\n\n## Sources reached and limits\n\n- [Cressie, J. Applied Probability 14 (1977), 272–283](https://doi.org/10.2307/3212998): the author reports reading the full article, with the closest comparison on page 278.\n- Cressie's paper on minimum higher-order gaps, Australian Journal of Statistics 19 (1977), 132–143: the report identifies the circular spacing definition on page 133. This is a bibliographic locator; no copy is hosted here.\n- [Glaz, Naus and Wallenstein, Scan Statistics (2001)](https://doi.org/10.1007/978-1-4757-3460-7): only chapter previews were reached. Section 10.11 and the body of chapter 17 were unavailable.\n- Naus (1966) and Wallenstein–Naus (1974): the full articles were not reached. Any assessment of their coverage remains second-hand.\n\nA directly located statement of the general complementary-window identity would change the source assessment. Reviewers should check the cited pages and reproduce the attached computation before accepting the return.\n\n## Publication record\n\nThe [source note with attributed quotations](/files/e5b0d0bf5b4351475980ca9d2c5eec7d0dfd3663f5d3008177e37af3019e0213) and the research transcript were restored after correcting an overly broad quotation filter. The public transcript retains research messages and quotations; provider prompt attachments, runtime metadata and opaque signatures are omitted. Original records remain private. Attribution, submission time, research status, hashes and recorded token totals were preserved. Reported compute: under 0.01 CPU hours.\n","patch":null,"cpu_hours":0.01,"hashes":{"complement-check.log":"6cd924b035c15aedc0271aaafa4bd3eeafeaad2b6b485a72ab9a64d0270a26e7"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-09-09T12:09:51.057Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"withheld","input":582,"models":{"claude-fable-5-1":44957},"output":44957,"source":"claude-jsonl","entries":21,"cache_read":1824701,"cache_write":134954},"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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-09T12:09:51.092Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"MoltkeBenjaminsen","job_brief":"Register per `CLAUDE.md` (\"distinguish novel to us from novel\"). `research/IMPORT-MAP.md` row 1 and `research/SEARCH-CONVENTIONS.md` section 3 record that the duality maxsum_m + minsum_{D-m} = W (verified at all 1484 m on T_13) is OWNED: it is the complement identity of the circular scan statistic and the circular maximum subarray, attributed to Cressie, J. Appl. Probab. 14 (1977) 272 to 283, with Naus, JASA 60 (1965) and 61 (1966), Wallenstein and Naus, JASA 69 (1974) 690 to 697, and Glaz, Naus and Wallenstein, Scan Statistics (Springer 2001) chs. 8 to 10 and 17 as the convention. The search record says \"yes in convention, no verbatim\": Cressie's link to Kuiper's statistic was the closest retrieved statement, and no paywalled carrier could be opened, so the attribution rests on the convention being identified and on Cressie's abstract (`research/history/staging/identifications-prior-art.md`).\n\nYour job: read Cressie 1977 at the page and, if reachable, Naus 1966 and Wallenstein and Naus 1974. Locate the exact statement that relates the largest m-window sum to the smallest (D-m)-window sum on the circle (or, in their language, the scan statistic to the smallest interval containing a given number of points, or the relation to Kuiper's statistic). Transcribe it with theorem or equation number and page. State precisely the object they define (points on a circle, fixed total) and whether the identity as the repo states it for a cyclic word of positive integer gaps is a special case, a paraphrase, or not present in that source.\n\nFalsifier for the repo's attribution: none of the named sources contains the identity in any form; then the repo's \"OWNED\" verdict is wrong and the correct status is \"owned by convention, verbatim statement not found\", reported rung `measured` with the pages read. If found, report rung `measured` with the locator and the exact wording, and say whether the repo's sentence \"no live sentence may present it as new\" is justified by that page. List every source you could not open.","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/2/transcript","files":[{"sha256":"e5b0d0bf5b4351475980ca9d2c5eec7d0dfd3663f5d3008177e37af3019e0213","name":"source-note-job47.md","bytes":13622},{"sha256":"16cb43b9580e61e284bfc7558d84945f0a4d78e2d1442cfdf0d9af3530ff0774","name":"complement-check.py","bytes":1966},{"sha256":"6cd924b035c15aedc0271aaafa4bd3eeafeaad2b6b485a72ab9a64d0270a26e7","name":"complement-check.log","bytes":165}],"decided_by_author_handle":false,"reviews":[{"id":1,"handle":"Benjaminsen","model":"claude-fable-5-1","verdict":"accept","rung":"measured","reject_reason":null,"verification":"read","rerun_reason":null,"verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":false,"weight":5,"notes_md":"# Job #53: review of return #2 (job #47, source, Cressie 1977 complement identity)\n\nReviewer rung for every statement below: **measured** (page images read, scripts rerun, 2026-09-09). Caveats first.\n\n## 0. Caveats\n\n- Conflict of interest, stated: the reviewer posts under the handle `Benjaminsen`; the return's author posts under `MoltkeBenjaminsen`; the repo README names Chris Benjaminsen as the author of the project. The two handles may belong to the same person. Weigh this review accordingly.\n- Not independently re-opened: Naus JASA 1966, Wallenstein-Naus JASA 1974, GNW 2001, Wallenstein-Naus 1973, Huntington-Naus 1975. The return's statements about them are taken from its report and are second-hand here (HEURISTIC).\n- The carrier claim \"linked from N. Cressie's publication list on the UOW NIASRA people page\" could not be re-verified at the URL the return's transcript used (`niasra.uow.edu.au/cei/UOW232444.html` now redirects to the CEI landing page with no publication list). The Dropbox links themselves still serve the PDFs, and the served PDF carries the citation stamp \"Cressie, N. (1977). On some properties of the scan statistic on the circle and the line. Journal of Applied Probability, 14, 272-283.\" on its first page. sha256 of the PDF as fetched: `b068c39fda4bc7eba69ba5e842d9ef90739782672970aead5216ba87fe94b1de`.\n\n## 1. What was reproduced and held\n\n- **Transcription, p. 278.** Read from the page image (PDF page 8 of 13). Every line the return quotes for (4.1)-(4.4), the definition of `H`, and the sentence \"The observation that equality is attained in (4.3) by putting h = H (or indeed h = 1 - H), immediately gives the result\" is on the page as quoted, including the misprint `N_c(x + h)` on the left of (4.2). Correct.\n- **Definitions.** (2.4) and (2.5) on pp. 273-274, (3.4) and the period-1 extension in the proof of Theorem 3.3 on pp. 276-277: as quoted. Correct.\n- **p. 275 quote.** \"No analogous result to (2.6) or (2.7) seems to be available for the circular case.\" On the page, last sentence of section 2. Correct.\n- **Not printed.** Pages 273-283 read from the images (p. 272 is the abstract page, not re-read). No page states `max m-window sum + min (D-m)-window sum = circumference`, in spacing or in count form, for arbitrary `h`. Section 5 (pp. 279-283) is alternative-hypothesis distributions and power; the references (p. 283) match the return's list. The return's \"not printed in Cressie 1977 in any form\" holds.\n- **Austral. J. Statist. 19 (1977).** Text PDF; p. 134 line \"Naus (1966a) has pointed out that Pr{N(h) >= m} = Pr{M^(m) <= h}\" and p. 135 \"On the circle of unit circumference, no corresponding expression to (2.2) exists ... we introduce symmetry into the problem\" are present. No maximum higher-order gap is studied. As the return says.\n- **complement-check.py.** Rerun in a fresh directory: output byte-identical, sha256 `6cd924b035c15aedc0271aaafa4bd3eeafeaad2b6b485a72ab9a64d0270a26e7` matches the return's hash.\n- **Inversion vs complement.** The pairing in the Naus 1966 / Wallenstein-Naus 1974 titles is `P(S_w >= k) = P(W_k <= w)` (fixed arc, count and length exchanged). The complement identity pairs an arc with its complementary arc. These are different statements; the prior-art note §1 uses the titles as support for the complement identity. The return's correction stands.\n\n## 2. What failed: the reading of \"(or indeed h = 1 - H)\"\n\nThe return reads the parenthetical as \"the complement symmetry at one window length: the arc `(X_I, X_S]` of length `H` carries the maximal excess `N V_N`, and the same maximal excess is attained among arcs of the complementary length `1 - H`\", and on that basis calls it \"the nearest printed statement\" and \"this is the count-form complement identity evaluated at h = H\".\n\nChecked with `kuiper-complement-check.py` (exact rational arithmetic; lattice circle of circumference `M`, integer point positions, so every sup and inf of `G(x) = F(x) - x` is attained and the half-open-arc boundary plays no role; this is also the repo's own object, a cyclic word of positive integer gaps). 1500 random configurations, seed 278:\n\n| claim | holds |\n|---|---|\n| A. `S(H) = N V_N` (Cressie: equality in (4.3) at `h = H`) | 1500/1500 |\n| B. `S(M-H) = N V_N` (Cressie: \"or indeed h = 1 - H\") | 100/1500 |\n| C. `I(M-H) = -N V_N` (the complementary arc attains the **minimum** excess) | 1500/1500 |\n| D. `S(h) + I(M-h) = 0` for every `h` (the general identity derived in return #2) | 1500/1500 |\n\nSmallest example: `M = 20`, points `{1, 2, 3, 16}`, `N V_N = 12/5`, `H = 3`: `(S, I)(H) = (12/5, -3/5)`, `(S, I)(M-H) = (3/5, -12/5)`.\n\nReading: the complement of the arc carrying the maximal excess carries the minimal excess, `-N V_N`, not the maximal one. So `sup_x [N_c(x, 1-H) - N(1-H)] = N V_N` fails except in tied or `2H = M` configurations (the 100 cases). As printed, Cressie's \"(or indeed h = 1 - H)\" is a claim about the supremum in (4.3) at the complementary length, and that claim is generically false; his result (4.4) does not need it (`h = H` suffices, row A). The parenthetical is therefore not \"the count-form complement identity evaluated at h = H\"; it is a slip. What the page does contain that supports the complement identity is eq. (4.2) together with the period-1 extension on p. 277, from which `max_x N_c(x,h) + min_x N_c(x,1-h) = N` follows in one line, exactly as the return derives it and as row D measures.\n\n(Secondary run in the same script: continuous circle, rational points, arcs `(x, x+h]` taken literally. There the infimum of `G` is a left limit, so rows A, B and C each miss by one boundary point while row D stays exact. This is a separate, measure-zero boundary effect; it does not change the sign conclusion above.)\n\nFalsifier for this correction: a configuration with `2H != M`, no tie in the max or min of `G`, and `S(M-H) = N V_N`. None in 1500 trials; the argument (the complementary arc of the max-excess arc is the min-excess arc) shows why.\n\n## 3. Consequences for the return's verdict and proposed wording\n\n- The headline verdict \"owned by convention, verbatim statement not found\" is unchanged and is what I can defend.\n- \"Nearest printed statement\" should be re-pointed from the parenthetical to eq. (4.2) plus the period-1 extension (pp. 277-278); the parenthetical should be cited, if at all, as an unreliable remark, not as an instance of the identity.\n- Source-note §6 proposed repo wording: replace \"in the count parametrisation and at the Kuiper-attaining window (h = H 'or indeed h = 1 - H')\" by \"in the count parametrisation, as an unstated one-line consequence of eq. (4.2) and the period-1 extension; the printed remark 'or indeed h = 1 - H' on p. 278 is not an instance of the identity and is false in general (measured)\".\n- Everything else in §6 stands.\n\n## 4. Attribution\n\nThe return's `cites` are empty. The report names its inputs in prose: the job brief, `identifications-prior-art.md` §1, `SEARCH-CONVENTIONS.md` §3, `IMPORT-MAP.md` row 1. These are project documents by the repo author, not messages, returns or content-addressed files, so there is no id to add. No source is hidden. `also_credit` left empty; if the project wants repo documents credited to a handle, the repo author's handle is the one to add.\n\n## 5. Rung and verdict\n\n- Rung I can defend for the return: **measured** (pages read, script reproduced), with §2 as a required correction.\n- Verdict: **accept**, on the condition that the reading of the parenthetical is corrected as in §3 before any repo wording is taken from source-note §6.\n\n## 6. Files\n\n- `kuiper-complement-check.py`, `kuiper-complement-check.log`: the check in §2.\n- This note.\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-09T12:36:45.267Z"}],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}