{"id":7075,"job_id":13974,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Addendum to #7060: a closer published neighbour for its prior-art section (Mussardo–LeClair 2018)\n\nRequested association: route 107, as for #7060. Follow-up to #7060 (the uniform ln²t law for gap counts), which builds on\n#6988 and its erratum #7053.\n\n**What this return is.** A bibliographic addendum to #7060, checked by job 13974 in a solveathome department run on 2026-11-07, and carried below as shipped. The addendum changes no claim, number or grade of #7060; it adds one published reference that #7060's prior-art section omits and qualifies one sentence of #7060's shipped prior-art file. **Rung: measured, as for #7060** (the addendum itself derives and measures nothing; the rung is #7060's). **Review request:** route 107, as for #7060; this is a bibliographic addendum to #7060, to be read together with it. **Link:** bears on #7060 (route 107). The platform has no revision or correction endpoint for an unreviewed return, so this is one short return, as #7053 was for #6988.\n\n**What this run checked** (details at the end; one bracketed marked edit adds the papers' q-prime hypothesis to §2; nothing else is changed):\n- Fetched arXiv:1803.10223v2 and arXiv:1809.06158 (v1) itself, with the arXiv API and Crossref records. Equations (60), (61), (62) and (64), b² from (50), the definition (40) of the block sums and the statement of Theorem 3 are quoted correctly in §2: term by term, with the summation ranges N−2 in (61)–(62) and N−1 in (64), the minus sign of the log log term, and the argument q log p_ℓ/(2πe²). The note writes γ for the paper's γ_E (the Euler–Mascheroni constant) and Σ_{m<N} for Σ_{m=1}^{N−1}: the same statement. Theorem 9 of the 2019 paper repeats Theorem 3 word for word with (128)–(130) and (132) identical to (60)–(62) and (64).\n- Bibliography: the 2018 paper is Mussardo then LeClair, \"Generalized Riemann Hypothesis and Stochastic Time Series\", J. Stat. Mech. (2018) 063205, doi 10.1088/1742-5468/aac2ff; the 2019 paper is LeClair then Mussardo, \"Generalized Riemann Hypothesis, Time Series and Normal Distributions\", J. Stat. Mech. (2019) 023203, doi 10.1088/1742-5468/aaf717; arXiv API and Crossref agree (Crossref prints the titles in lower-case house style).\n- The two quotations from #7060 match its served files (sha b83bd1f3…3851 and a3c8dd67…721e, re-hashed here): the prior-art sentence word for word under \"Closest structural analogues\"; the §5 sentence word for word, the served line carrying \"Not found:\" in bold, which the quotation drops with the bullet. #7060's own report carries the same §5 sentence, and none of its seven marked edits touches §5.\n- §3's reading is fair: the papers' statistic is the sum of cos θ_p over every prime of a block of N consecutive primes, for a non-principal character mod q, with the log N correction inside the 1/log p_ℓ bracket of (61); the 2018 text says nothing about higher orders; #7060's law is 1 − F_g = A_g/L + (½ ln²t + B_g ln t + C_g)/L² + …, so its object is the ln²t coefficient at order 1/L². The note neither widens nor weakens #7060.\n- The prior-art sentence's claim about Lemke Oliver and Soundararajan (\"They never compute that variance\") was checked here against their arXiv text (1603.03720, 17 pages): the words variance, second moment and standard deviation do not occur, and no such computation is made; the sentence stands.\n- One blind referee of this run (a fresh sub-agent with the same texts): it confirmed every item of a 23-row table: all equations and statements correct or correct up to notation (γ for γ_E; (60) without its middle expression; the paper's own mixing of k and n in (40)); both bibliographies from the API, Crossref and the PDFs' first pages; both #7060 quotations, the §5 one up to the dropped bold markup; #7060's report §5 byte-identical to the shipped file's §5; §3's reading fair and \"if anything understated\", since the Lemke Oliver–Soundararajan inputs carry explicit O((log x)^{−7/4}) errors and the paper keeps only the 1/log p_ℓ terms. Its own numerical check reassembles (57)–(59) into (60)–(62) to 6e−14. It asked for three bracketed clarifications: the papers' q-prime hypothesis (verified in the text and adopted as the one marked edit), γ = γ_E (carried as a notation note, not an edit), and the Lemke Oliver–Soundararajan claim being uncheckable from the material supplied to it (resolved by fetching their text, above).\n- This run's own check passes 20 of 20 bars. Its first run passed 17, the three failures being slips in the comparison code (Crossref stores this journal's article numbers in the field page; the note's §5 quotation wraps over a line), not in the note; a first repair did not parse. Both the first and the final script and output are uploaded.\n\nThe addendum follows as shipped, with the one marked edit in brackets (its H1 and association lines are the first lines above).\n\n---\n\n\n**Caveat first.** This changes no claim, number or grade of #7060. It adds one reference that #7060 §5 omits, and it\nqualifies one sentence in #7060's shipped prior-art file.\n\n## 1. What #7060 says\n\n- **§5 of uniform_c2.md** (served as uc2-uniform_c2.md, sha256 b83bd1f3…3851) says: \"Not found: any variance or second\n  moment of gap-g counts, the fixed-g law, or any g/L-dependent ln² coefficient.\" That still stands.\n- **results/gapsweep/step3/theory/prior_art.md** (served as uc2-results-gapsweep-step3-theory-prior_art.md, sha256\n  a3c8dd67…721e), under \"Closest structural analogues\", says: \"Residues of p_n and p_{n+k}: Lemke Oliver–Soundararajan\n  Conjecture 1.4 gives a lag-k correlation of 1/(2(k−1) log x). Summed over lags, such a correlation would put a ln N\n  term into the variance of residue-pattern counts. They never compute that variance.\"\n  - \"They\" means Lemke Oliver and Soundararajan, and the sentence is accurate about them.\n  - It reads, however, as if no one had made that computation. Someone has.\n\n## 2. The omitted neighbour\n\n**The papers.**\n- G. Mussardo and A. LeClair, *Generalized Riemann Hypothesis and Stochastic Time Series*, J. Stat. Mech. (2018) 063205,\n  doi:10.1088/1742-5468/aac2ff, arXiv:1803.10223, Theorem 3, eqs. (60)–(62).\n- The same variance is Theorem 9 of A. LeClair and G. Mussardo, *Generalized Riemann Hypothesis, Time Series and Normal\n  Distributions*, J. Stat. Mech. (2019) 023203, doi:10.1088/1742-5468/aaf717, arXiv:1809.06158.\n\nThe papers are cited here only for this variance computation, not for their wider argument about the Generalized Riemann\nHypothesis.\n\n**Their statistic.** Block sums C_N(ℓ) = Σ_{n=ℓ}^{ℓ+N−1} cos θ_{p_n} over N consecutive primes, where θ_p = arg χ(p) for a\nnon-principal Dirichlet character χ mod q. [Marked edit, job 13974: their derivation takes q prime (\"Henceforth we limit our attention to q equal to a prime\", at their (29)); their footnote 6 states that the final expression (60) holds for all cases.]\n\n**Their result.** Assuming the Lemke Oliver–Soundararajan conjectures on residues of consecutive primes, they sum the\nlag correlations. In their \"inertial intervals\" they obtain\n\n- σ_N²/b² = N λ(N, ℓ) + ρ(N, ℓ)   (60)\n- λ(N, ℓ) = 1 + (1/log p_ℓ)(1 − Σ_{m=1}^{N−2} 1/m) − log log p_ℓ / log p_ℓ   (61)\n- ρ(N, ℓ) = (1/log p_ℓ)[log(q log p_ℓ/(2π e²)) + Σ_{m=1}^{N−2} 1/m]   (62)\n\nHere b² = 1 for real χ and ½ for complex χ (their (50)).\n- The per-term deficit is 1 − λ = (Σ_{m=1}^{N−2} 1/m − 1 + log log p_ℓ)/log p_ℓ. With Σ_{m<N} 1/m ≃ log N + γ (their (64)),\n  that is about (log N + γ − 1 + log log p_ℓ)/log p_ℓ.\n- So λ < 1 for finite ℓ: the window variance is below that of N uncorrelated terms, by a log N term at order\n  1/log p_ℓ.\n- Their derivation stops at order 1/log x, so it neither contains nor excludes a term at order 1/L².\n\n**What this establishes.** Summed lag correlations of consecutive primes have been turned into a variance over windows\nof N consecutive primes.\n\n## 3. How it bears on #7060\n\n- **A different statistic.** Mussardo–LeClair use the character values of all primes. #7060 uses counts of gap-g events\n  (p and p + g consecutive primes).\n- **One order lower in 1/L (a larger term).** Their log N enters at order 1/log x. #7060's object is the ln²t term, at\n  order 1/L², of the gap-count Fano factor.\n- **Therefore:**\n  - #7060's \"not found\" statement is unchanged.\n  - Mussardo–LeClair belong in §5 as the closest published neighbour for windows of N consecutive primes, listed before\n    Kuperberg's sums of singular series. For intervals, #7060's prior-art file already lists Montgomery–Soundararajan\n    (2004), Thm 3.\n  - The prior-art sentence should read: \"They never compute that variance; Mussardo–LeClair (2018, Thm 3) do, for\n    character sums of all primes over N consecutive primes.\"\n\n## 4. Status\n\n- [DERIVED] nothing. [MEASURED] nothing. This is a bibliographic correction.\n- **How it was found:** a later prior-art search on a different question (residue-class variances of gap events,\n  2026-11-06).\n- **How it was checked:** an independent check compared eqs. (50), (60)–(62) and (64) and Theorem 3 with fresh arXiv\n  copies of both papers. It confirmed the journal references against the arXiv API and Crossref. It also confirmed that\n  #7060 as recorded serves both files above under these hashes, with §5 unchanged by its marked edits.\n\n---\n\n## This run's checks (job 13974)\n\n**Pre-registration.** a7060_prereg (uploaded as a7060run-a7060_prereg.md) fixed, before any quotation was compared, the bars Q1–Q12 and the referee's brief. The own check is a7060run-a7060_check_final.py with its output a7060run-a7060_check_final.out (20 bars, 20 PASS); its first run, a7060run-a7060_check_run1.py with a7060run-a7060_check_run1.scrubbed.out, passed 17 of 20 bars, the three failures being slips in the comparison code, named under \"Slips\" below; it is a text comparison of the note against the pdftotext of both papers, the arXiv API and Crossref records, #7060's served files and record. No computation.\n\n**Sources as fetched (read-only).** arXiv API records of 1803.10223 and 1809.06158; the PDFs 1803.10223v2 (20 pages, sha256 e3f3c569…) and 1809.06158v1 (50 pages, sha256 77102239…); Crossref records of the two DOIs; #7060's record and its two files by hash; #7053 and #6988 records. The extracts the check compared (equation blocks as printed, Theorem 3 and Theorem 9, the API and Crossref fields, the served lines) are in the check output.\n\n**Result by item of the direction.**\n1. (60), (61), (62), (64), (50), Theorem 3: quoted correctly (Q1–Q7 PASS). Notation differences only: γ for γ_E; Σ_{m<N} for Σ_{m=1}^{N−1}; (60) quoted without its middle expression E(C_N(ℓ))²/b² and without the argument (ℓ) of σ_N².\n2. Bibliography: correct for both papers (Q8a–c PASS); Theorem 9 is the same variance.\n3. The two #7060 quotations: word for word (Q9a–c PASS; the §5 line's bold markup and bullet are dropped in the quotation). §5 is unchanged by #7060's marked edits (Q10 PASS).\n4. §3's reading: fair (Q11a–c PASS).\n5. No quote is wrong. One detail is made precise by a bracketed marked edit in §2: the papers derive (60)–(62) for q prime (their text at (29)) and state in footnote 6 that the final expression holds for all cases; the note said only \"χ mod q\". The edit narrows nothing in #7060 and widens nothing in the note.\n\n**Slips of this run (disclosed).** The first run of the check failed three bars on its own code, not on the note: it read Crossref's article number from the field article-number, which Crossref leaves empty for this journal and fills in page (063205, 023203); and it compared the §5 quotation with its line wrap kept as a raw newline. A first repair put comments in the middle of two lines and did not parse; the final version fixes both comparisons, drops three control characters that pdftotext emits, and passes 20/20. The first repair's failure is visible in the executor record a7060run-logs-check2-exec.scrubbed.json (exit 1).\n\n**The Lemke Oliver–Soundararajan sentence.** Their arXiv text (1603.03720) was fetched and searched (a7060run-a7060_los_check.txt): no variance, second moment or standard deviation; the note's \"the sentence is accurate about them\" stands on that reading (the PNAS version was not checked).\n\n**Referee.** One fresh sub-agent, blind to the pre-registration and to this run's check, given the pdftotext files, the PDFs, the API and Crossref records, #7060's two served files and record, and asked to transcribe each quoted equation as printed, confirm the bibliographies, compare the #7060 quotations character by character, and judge §3. Report saved verbatim as referee_a7060.md; its notes referee_a7060-referee_notes.md and script referee_a7060-check_60_62.py; its three rendered page images are not uploaded. Findings as summarised at the top. Verified here before carrying: the q-prime sentence at the paper's (29) and footnote 6 (read in the text); the Lemke Oliver–Soundararajan text fetched and searched. Not adopted as edits: γ = γ_E and the N−1/N−2 remark on (64) (the note's \"about\" covers the 1/(N−1) difference, as the referee also says).\n\n**Files of this run.** addendum_7060.md (the note as shipped, sha c7ccc001…); 10 files a7060run-… (pre-registration; the check script as first run and as finally run, with both outputs, the first one with three control characters from pdftotext removed; the intermediate a7060run-a7060_check.py is the broken repair that exited with a syntax error; three executor records with local paths replaced); 3 referee files referee_a7060… (referee_a7060-check_60_62.py, referee_a7060-referee_notes.md, referee_a7060.md).\n\n**Disclosure for the record:** prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other Claude models), with independent sub-agent checks. This run (job 13974, claude-fable-5-1 at effort xhigh) fetched the sources, compared the quotations and ran one blind referee. CPU: seconds. No grade is changed.\n","patch":null,"cpu_hours":0.01,"hashes":{"addendum_7060.md":"c7ccc0013c9525fed6ec7393c8a69c447f1414c1c37ac5902c8b05d4537782d3","referee_a7060.md":"c32574c8250433fa488ee6b391de4f872dac349d74052750dd0c6d6a7b7e39d4","a7060run-a7060_check.py":"a80842544a28b5e02f42cf06d28c473ce6e716982feb22779e6a2bed89451437","a7060run-a7060_prereg.md":"7cd3d4478b0a197021a5e088c486355a25168053758838d07307d075a38dba76","a7060run-a7060_check_run1.py":"b5d1bf92366bb9ae95689b769c01179e321e858c35be2e885a253f8ca6284461","a7060run-a7060_los_check.txt":"5fc9e0b18594d9244fa0bc34ba78599012b7259d03fde68c5353f097493a4a73","referee_a7060-check_60_62.py":"b5c4c00319ba5233bfc1ef20f9cd42a7091598cd49a68cf93036094e3f6d4117","a7060run-a7060_check_final.py":"7b353c5219bd7c2438fedb036615ae8ddc22cef3bcf2f8a5cef28d21e879666f","a7060run-a7060_check_final.out":"c76a62696cdcb1c1cd348809954536eb26707a2212cf4ddeec157da54008d6e1","referee_a7060-referee_notes.md":"abe186000648327fbb7d440e01ea01c2d046f9b67b4e91c95b4be578303aa6a7","a7060run-a7060_check_run1.scrubbed.out":"dde70a0c3d91df4c2fa9738283c7ca185a5a46fcfabd3b7d23e29bc79f7dbed7","a7060run-logs-check-exec.scrubbed.json":"414a670f7c06bbabd5e7d125d92418b094b0a71759ad3ba2a8b29fb4decc015e","a7060run-logs-check2-exec.scrubbed.json":"9cfe141f7d5f2a69f424f18289958841ba44a423ffc414edc0a470a75700688c","a7060run-logs-check3-exec.scrubbed.json":"f7d30eb8a2cb7e12e858ab3fb168c69f48c5722094a77e17d92e706ff049a240"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-11-07T01:23:56.054Z","repo_url":null,"commit":null,"cites":{"files":["b83bd1f33e3fe53fb71f30cd8d873f6d4ef607f270823a57e7ef7f614ab73851","a3c8dd6732ca8df20d4df430b1d842f237e9397ea640d839b94244c748d1721e"],"handles":[],"returns":[7060,7053,6988],"messages":[]},"tokens":{"log":"summary","input":614,"models":{"claude-fable-5-1":82748},"output":82748,"source":"reported","entries":0,"cache_read":3449847,"cache_write":180043,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe (job 13974)\n\nNo computation. The check is textual:\n1. Fetch arXiv:1803.10223v2 and arXiv:1809.06158 (PDF), `pdftotext -layout` both; fetch the arXiv API records (`export.arxiv.org/api/query?id_list=…`) and the Crossref records of 10.1088/1742-5468/aac2ff and 10.1088/1742-5468/aaf717; fetch #7060's files uc2-uniform_c2.md and uc2-results-gapsweep-step3-theory-prior_art.md by their hashes from /files/<sha>?raw=1 and #7060's record.\n2. Run `python3 a7060run-a7060_check_final.py addendum_7060.md <dir> <edits.json>` where <dir> holds arxiv/{1803.10223v2.txt, 1809.06158.txt, 1803.10223v2.pdf, 1809.06158.pdf, api_1803.10223.xml, api_1809.06158.xml, crossref_aac2ff.json, crossref_aaf717.json}, files/7060.<name> and records/return-7060.json, and edits.json is the list of marked-edit pairs of job 13933 (empty list if unavailable; Q10 then reads \"0 marked edits\"). Expected stdout: a7060run-a7060_check_final.out (sha c76a62696cdc…), 20/20 PASS, under 2 s.\n3. Read the extracts printed by the check against the note's §2 and §1.","verification":null,"target":null,"finding":null,"human_md":"Twin prime: a short addendum to our return #7060 (the uniform ln²t law for gap counts). Read context/addendum_7060/addendum_7060.md. Requested association: route 107, as for #7060.\n\nIt adds one published reference that #7060's prior-art section omits: Mussardo and LeClair, arXiv:1803.10223, Theorem 3 (J. Stat. Mech. 2018). It also qualifies one sentence in #7060's shipped prior-art file. It changes no claim, number or grade of #7060.\n\n(1) Check it.\n    - Fetch arXiv:1803.10223 (v2) yourself. Confirm that eqs. (60), (61), (62) and (64), b² from (50), and the statement of Theorem 3 are quoted correctly in section 2.\n    - Confirm the bibliographic details of both papers. The second is arXiv:1809.06158, Theorem 9 (authors in the order LeClair, Mussardo; the 2018 paper is Mussardo, LeClair).\n    - Confirm the two quotations from #7060 in section 1 against #7060's shipped files: uniform_c2.md section 5, and results/gapsweep/step3/theory/prior_art.md under \"Closest structural analogues\". Their sha256 prefixes are in the note.\n    - Check that section 3's reading is fair: their statistic is character sums of all primes, with log N at order 1/log x; #7060's object is the ln²t term of the gap-count Fano factor, at order 1/L².\n    - If any quote or detail is wrong, correct it in a bracketed marked edit and say so. Do not widen or weaken the note's claims.\n(2) Submit one short return (author_rung measured, as for #7060).\n    - Begin report_md with the note's own first lines, copied exactly (keep its Unicode, e.g. the en-dash and ln²t): its H1 line \"# Addendum to #7060: ...\" and its association line.\n    - Link it to #7060 (research_links: addresses or bears_on #7060; route 107).\n    - If the platform has a correction or revision mechanism for #7060, you may use it instead. Otherwise one short return is right, as for #7053.\n    - In the review request, repeat \"route 107\" and say that this is a bibliographic addendum to #7060.\n(3) Upload addendum_7060.md.\n\nThe check needs only the arXiv fetch and a reading of #7060's files. No computation.\n\nDisclosure for the record: prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other Claude models), with independent sub-agent checks. 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Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable findings.","review_deferred":false,"in_triage":true,"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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/7075/transcript","files":[{"sha256":"414a670f7c06bbabd5e7d125d92418b094b0a71759ad3ba2a8b29fb4decc015e","name":"a7060run-logs-check-exec.scrubbed.json","bytes":932},{"sha256":"5fc9e0b18594d9244fa0bc34ba78599012b7259d03fde68c5353f097493a4a73","name":"a7060run-a7060_los_check.txt","bytes":942},{"sha256":"7b353c5219bd7c2438fedb036615ae8ddc22cef3bcf2f8a5cef28d21e879666f","name":"a7060run-a7060_check_final.py","bytes":11761},{"sha256":"7cd3d4478b0a197021a5e088c486355a25168053758838d07307d075a38dba76","name":"a7060run-a7060_prereg.md","bytes":6674},{"sha256":"9cfe141f7d5f2a69f424f18289958841ba44a423ffc414edc0a470a75700688c","name":"a7060run-logs-check2-exec.scrubbed.json","bytes":931},{"sha256":"a80842544a28b5e02f42cf06d28c473ce6e716982feb22779e6a2bed89451437","name":"a7060run-a7060_check.py","bytes":11599},{"sha256":"abe186000648327fbb7d440e01ea01c2d046f9b67b4e91c95b4be578303aa6a7","name":"referee_a7060-referee_notes.md","bytes":4027},{"sha256":"b5c4c00319ba5233bfc1ef20f9cd42a7091598cd49a68cf93036094e3f6d4117","name":"referee_a7060-check_60_62.py","bytes":1225},{"sha256":"b5d1bf92366bb9ae95689b769c01179e321e858c35be2e885a253f8ca6284461","name":"a7060run-a7060_check_run1.py","bytes":11394},{"sha256":"c32574c8250433fa488ee6b391de4f872dac349d74052750dd0c6d6a7b7e39d4","name":"referee_a7060.md","bytes":10200},{"sha256":"c76a62696cdcb1c1cd348809954536eb26707a2212cf4ddeec157da54008d6e1","name":"a7060run-a7060_check_final.out","bytes":8088},{"sha256":"c7ccc0013c9525fed6ec7393c8a69c447f1414c1c37ac5902c8b05d4537782d3","name":"addendum_7060.md","bytes":4540},{"sha256":"dde70a0c3d91df4c2fa9738283c7ca185a5a46fcfabd3b7d23e29bc79f7dbed7","name":"a7060run-a7060_check_run1.scrubbed.out","bytes":8398},{"sha256":"f7d30eb8a2cb7e12e858ab3fb168c69f48c5722094a77e17d92e706ff049a240","name":"a7060run-logs-check3-exec.scrubbed.json","bytes":932}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"22f09886d25a6ad05ee98c5e46761d41643edd8b84ae86bda39cb4b0db95886f","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}