{"id":5989,"job_id":4415,"problem_id":1,"lane_id":null,"type":"paper","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 4415: TeX notation for \"An upper bound for the twin Jacobsthal function\" (beta2-note)\n\nNotation-only revision. No word, number, claim, citation, heading, link, code span or path was changed.\n\n- **Started from:** the served manuscript, sha256 `75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67` (equals `current_file_sha`). No other revision was in review (`in_review` 0).\n- **Revised manuscript:** `beta2-note.md`, sha256 `2887fe4beac3905e3ecd947aa686342291f45024a551e0f6049ef479c4c35b60`.\n- **Diff:** `beta2-note-tex.diff` (unified, 14 hunks), against the version above.\n\n## What changed\n- **Formulas converted.** Every formula on the 62 lines that carried bare `^{…}`/`_{…}` is now in `$…$` or `$$…$$`. The 58 ordered replacements are listed in `typeset.js` (uploaded).\n- **Display formulas.** The set-apart formulas in §2–§3 are now `$$…$$` blocks: the Ω(κ,L) product condition, the Theorem 9.1 lower bound, ω(d), (∗), the dimension product, V(z), and the remainder chain (as an `aligned` block).\n- **TeX symbols inside formulas:** `\\kappa`, `\\omega`, `\\nu`, `\\varepsilon`, `\\beta_2`, `p_n`, `\\prod`, `\\sum`, `\\mu^2`, `\\tau_8`, `\\ll_\\varepsilon`, `\\asymp`, `\\sim`, `\\le`/`\\ge`, `\\log`, `\\ln`, `\\mid`.\n- **Bold kept outside math.** The theorem line `**G₂(n) ≤ C(ε) · pₙ^{β₂+ε}**` keeps its `**` outside the math.\n- **Line breaks.** Two formulas that were broken across lines (lines 20–21 and 209–210) are written as one or two complete inline formulas. That is a whitespace change only.\n\n## Checks (`typeset.js`)\n- **Text outside math unchanged.** Removing every `$…$`/`$$…$$` span gives exactly the original text with the replaced formula strings removed (whitespace-normalised).\n- **No residual raw TeX.** No bare `^{` or `_{` remains outside math.\n- **No stray delimiters.** No `$` remains outside the math spans, and the original contained no literal `$`.\n- **Exhaustive conversion.** All 58 patterns matched.\n\n## Scope note\n- **Unicode-only formulas.** Formulas written only in Unicode with no raw TeX syntax (for example `gcd(r(r+2), pₙ#) = 1`, `G₂ = 2, 6, …`) were left as rendered Unicode, as in the anchored-note revision (return #5986).\n- **Not checked visually.** I did not render the page in a browser.\n\n## Content defect noticed (not changed)\n- In §2, the status block (line 13) and §6.1 say the remainder weight is `2·4^{ν(m)}` \"verified\". §3's chain then bounds `|r_m| ≤ 2^{ν(m)}`, while (∗) states `|r_d| ≤ ω(d) ≤ 2^{ν(d)}` with ω(2) = 1. This is consistent. I note it only because ν counts all prime factors while ν′ counts odd ones. No change was made.\n","patch":null,"cpu_hours":0,"hashes":{"beta2-note.md":"2887fe4beac3905e3ecd947aa686342291f45024a551e0f6049ef479c4c35b60","beta2-note-tex.diff":"df152c9a424c125e7a11e442e6ce2efc74f5fb1e7517bce4dbabc806d962abea"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-26T08:29:12.087Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5986],"messages":[]},"tokens":{"log":"summary","input":16,"models":{"claude-opus-5-5":21288},"output":21288,"source":"reported","entries":0,"cache_read":3920963,"cache_write":40558,"observed_models":[]},"paper_slug":"beta2-note","revision_path":"paper/beta2-note.md","revision_sha":"2887fe4beac3905e3ecd947aa686342291f45024a551e0f6049ef479c4c35b60","recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","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":"2026-10-26T08:29:12.087Z","department_id":"dept_70682f6fbf6580733b92340d","run_id":"run_916812d086720b070e62de98","triage_lead":null,"revision_base_sha":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67","integration":null,"resolves":[],"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":"sarenayu","job_brief":"paper.slug: beta2-note\n\nTypeset the mathematics of \"An upper bound for the twin Jacobsthal function\" in TeX. Take the current manuscript from `GET /projects/twin-primes/papers/beta2-note` (JSON, `manuscript_md`). It writes math as plain text, with bare ^{…} and _{…} on lines 12, 13, 15, 45, 51, 62, 69, 120, 123, 167, 172, 173, 175, 178, 179, 189, 194, 198, 201, 202 and 42 more. The site typesets only $…$, $$…$$, \\(…\\) and \\[…\\], so those formulas show as raw text.\n\nChange the notation and nothing else: put each inline formula in $…$ and each formula set apart on its own indented lines in a $$…$$ block; write sub- and superscripts, Greek letters and operators inside a formula in TeX (`p_n^{\\beta_2+\\varepsilon}`, `\\sum_{m \\mid P(z)}`, `\\ll_\\varepsilon`). Leave every word, number, claim, citation, heading, link, code span and file path as it is, and write a literal dollar as `\\$`. A defect you notice in the content goes in your report, not into this revision. If another revision of this paper is under review, start from the current version all the same and name that return in your report.\n\nReturn the complete manuscript as one uploaded Markdown file with `\"paper\": { \"slug\": \"beta2-note\", \"file\": \"<sha256 of the manuscript>\" }`, and upload a diff against the version you started from (name its sha256 in the report), so a reviewer can check that only notation changed and that the page at /projects/twin-primes/papers/beta2-note renders every formula.","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":[{"id":5991,"handle":"sarenayu","status":"pending"},{"id":5997,"handle":"sarenayu","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5989/transcript","files":[{"sha256":"2887fe4beac3905e3ecd947aa686342291f45024a551e0f6049ef479c4c35b60","name":"beta2-note.md","bytes":31222},{"sha256":"df152c9a424c125e7a11e442e6ce2efc74f5fb1e7517bce4dbabc806d962abea","name":"beta2-note-tex.diff","bytes":22194},{"sha256":"b569eb597ebd6c387a5867a8b1ef9d65002b8f75b2a63daabb3f32286e5503ad","name":"typeset.js","bytes":8022}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9edda0ba603d17ad61fdc26757acab67be9090556637bfaad7f5bf2f33d7b441","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}