{"id":1663,"job_id":3073,"problem_id":1,"lane_id":null,"type":"audit","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# run-2026-09-25-r — job 3073 (audit): findings #202, #203, #215 in `paper/kk-lower-bound.md`; #55 checked and left open\n\n**What changed (five edits, three findings).** In the served text, whose base pinned here is\n`X-Content-SHA256` `8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce` (the\n`content_sha` of #202/#203/#215 and the served version 2 per\n`/history/paper/kk-lower-bound.md`):\n\n- **#202** — reference **[R4]** no longer points at the live path `/docs/paper/kk-lower-bound.md`\n  (which now serves this very revision, so R4's `§9`, `(5.1)`, `§6.3`, `§11.2` pointers would have\n  resolved into the wrong text). It is pinned to the prior edition at\n  `/projects/twin-primes/history/paper/kk-lower-bound.md`, labelled version 1, SHA-256\n  `7c375d95…107f` unchanged.\n- **#203(a)** — §3 Input H: the printed range is now quoted in the source's own notation\n  ($x=X$, $y=Z$): $y\\geq(\\log x)^{1+\\varepsilon}$. The stated $u\\leq Z^{1-\\varepsilon}$ implies it,\n  so §6.3 is unaffected (both conditions hold at the $\\varepsilon=1/2$ used there).\n- **#203(b)** — §9: the sentence asking the next review to check \"its uniformity for these changing\n  residue systems\" is now the reviewer's correction: uniformity is not a separate obligation,\n  because the quoted (7) carries a constant depending only on $\\kappa$ (in the printed theorem on\n  $\\kappa,A_1,A_2$), and here $\\kappa=4$ with $g(p)\\leq4$, $g(p)/p\\leq4/5$ at every $p\\leq z$ by\n  (18), so one constant is uniform along the construction.\n- **#215(a)** — §3 Input S plus a new reference **[FI]**: Friedlander–Iwaniec, *Opera de Cribro*\n  (AMS Colloq. 57, 2010), Thm 6.9 with Cor 6.10, pp. 68–69, added as a second carrier at\n  **OCR custody** (text extraction, explicitly not a page reading; pp. 68–69 still to be read at\n  page image).\n- **#215(b)** — §8: the priced reading is recorded — $C=2C_2e^{-2\\gamma}C_1=0.41621\\,C_1$,\n  $C_1(1/2)=805.5$ at $\\delta=0.05$, $K=3$, hence $c_0<3.7\\cdot10^{-4}$, conditional on that\n  reading — with the note (as the reviewer requires) that the value $7.5\\cdot10^{-4}$ recorded\n  elsewhere drops the factor at $p=2$ and is not used. The section still does not price $C$ as its\n  own argument.\n\nNo theorem, constant of the composition, grade or other byte changed. Revised file sha256\n`15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7` (bytes 27396; base 25748 B); diff `kk-lower-bound.patch` (`d2f5f79e07ffe4d682314cf72fac1df2d85f6e29d3ea08419f6577316e60cafa`), 5 hunks.\nUploaded `POST /files` → `ok:true`.\n\n## Finding #55: checked, and **left open** — its targets are not in the served edition\n\nFinding #55 is the `also_fix` block of review #261 of return #1538, whose `content_sha` is\n`7c375d95…107f` — the **version 1** text, i.e. the v1-based line that #1538 proposes to file\n(`kk-lower-bound.rev2.md`, 93045 B, sha `d1f4661b…`; its patch against the served baseline\n`7c375d95…` is 856 lines). Its three items are wording items for **that** filing:\n\n1. Appendix A row \"certified 479,339 at y = 5003 | §10\" → 479,340;\n2. §10: drop \"a referee is entitled to call the theorem unfalsifiable … That is accurate\";\n3. §11.4 rows \"the ledger's O(1)\" and \"the exponent assembly\": restate the falsifiers.\n\nNone of those passages is in the served edition. Checked this run on the fetched bytes: the served\ntext has no Appendix A or Appendix B, no section 11 or §11.4 (it ends at §10 plus the disclosure\nblock), no numeric row containing 479,339 (or any 479,3xx), no \"not exhibited\" / \"unfalsifiable\"\nsentence, and no \"assembly\" / \"ledger's O(1)\" rows; `grep -c '\\`\\`\\`'` = 0, so it has no embedded\nscript, recipe or generated block to run. Adding those passages would be inventing content in a\ndifferent manuscript, not the smallest complete correction. **#55 therefore stays open against its\nown `content_sha`** (`7c375d95…`) and is answerable only by the filing that carries that text\n(#1538's line), which is not this job's artifact.\n\nIts fourth remark is confirmed for the served text: the served file hashes to its served value **as\nfetched**, with a single trailing newline and nothing appended to strip — so any recipe step that\nstrips one appended newline before hashing is stale for this edition too. No recipe for this slug is\nserved at `docs/paper/kk-lower-bound.recipe.md`, `docs/paper/recipes/kk-lower-bound.md` or\n`docs/paper/kk-lower-bound.rev2.md` (all 404 this run), so there is nothing here to correct.\n\nThe relevant substance of item 3 is already carried by the served text in its own words: §6.2 says\nthe comparisons \"are comparisons up to constant factors, not identities with unit constant. In\nparticular, exponentiating an $O(1)$ in (20) does not remove it.\"\n\n## Scope and unresolved obligations\n\n- `resolves`: **[202, 203, 215]** only. #55 is reported above and remains open; no other finding on\n  this path is claimed.\n- The change is prose only: no execution, no computation, `cpu_hours` 0. The \"still runs / stdout\n  reproduces byte for byte\" clause is vacuous for this artifact (no executable, no OUTPUT block).\n- Embedded hashes: the file embeds SHAs of *other* documents ([R1]–[R7], the arXiv and HT PDFs);\n  none of those targets changed, so nothing needed re-embedding. The [R4] SHA stays `7c375d95…`\n  because its target is now explicitly the pinned version 1.\n- Observed but **not** fixed (out of scope, no finding listed for it): [R5]'s recorded SHA\n  `61601140…` for `research/SEARCH-CONVENTIONS.md` is older than that file's current revision\n  (`56757e0b…`), a pre-existing drift this job does not own.\n\n## Return fields\n\n`revision`: `{path: paper/kk-lower-bound.md, file: 15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7, base: 8ad20080…db84ce}`, `files`:\n[that sha], `resolves`: [202, 203, 215], `cites.returns`: [1538], `author_rung`: verified.\n\nTranscript: scrubbed export of this session's `log.jsonl` (`transcript.clean.jsonl`); the raw\nexport carries the account token and was scrubbed before embedding. Usage stays **pending** — this\napplication exposes no per-assignment token counts.\n","patch":"--- a/paper/kk-lower-bound.md\n+++ b/paper/kk-lower-bound.md\n@@ -145,7 +145,12 @@\n The statement was read in arXiv v2, page 4, and in the journal's online\n text. Its proof cites Halberstam and Richert, *Sieve Methods* (1974),\n Theorem 2.2. That printed book theorem has not been independently read\n-here. Sections 4--8 establish deductions conditional on using (7) in\n+here. A second carrier of the same statement is recorded at **OCR\n+custody**: Friedlander and Iwaniec, *Opera de Cribro* (AMS Colloquium\n+Publications 57, 2010), Theorem 6.9 with Corollary 6.10, pp. 68--69\n+[FI]. That custody is a text extraction, not a page reading; the two\n+pages remain to be read at page image, and this citation is not counted\n+as one. Sections 4--8 establish deductions conditional on using (7) in\n this stated form. They do not reconstruct its underlying sieve proof.\n \n **Input M.** Mertens' prime harmonic estimate is\n@@ -166,6 +171,9 @@\n  \\tag{9}\n $$\n as $Z,u\\to\\infty$, uniformly when $u\\leq Z^{1-\\varepsilon}$.\n+In the source's notation, $x=X$ and $y=Z$, the printed range is\n+$y\\geq(\\log x)^{1+\\varepsilon}$; the stated $u\\leq Z^{1-\\varepsilon}$\n+implies it, so the application in section 6.3 is unaffected.\n The formula and its range were read from the primary page image.\n We will use only its upper-bound consequence.\n \n@@ -454,6 +462,13 @@\n $\\pi(y)-\\pi(z)\\sim y/\\log y$. Choosing $c<1/(8C)$ leaves more\n than one unused prime per survivor, so the assignment in section 7\n finishes the cover.\n+\n+We do not price $C$ here. If the local factors are priced, the\n+project's reading of this input gives\n+$C=2C_2e^{-2\\gamma}C_1=0.41621\\,C_1$ with $C_1(1/2)=805.5$ at\n+$\\delta=0.05$, $K=3$, hence $c_0<3.7\\cdot10^{-4}$ conditional on that\n+reading. The value $7.5\\cdot10^{-4}$ recorded elsewhere drops the\n+factor at $p=2$ and is not used here.\n \n This argument does not use the smooth-number input, but it still uses\n Input S. It cannot be described as having escaped the unread\n@@ -490,8 +505,12 @@\n historical evidence, not executions by this manuscript's preparer.\n \n The next mathematical review should check Input S against a printed\n-upper-sieve theorem, including its uniformity for these changing\n-residue systems, and then inspect the reduction in sections 4--7.\n+upper-sieve theorem and then inspect the reduction in sections 4--7.\n+Uniformity over these changing residue systems is not a separate\n+obligation: in the quoted form (7) the constant depends only on\n+$\\kappa$ (in the printed theorem on $\\kappa,A_1,A_2$), and here\n+$\\kappa=4$ with $g(p)\\leq4$ and $g(p)/p\\leq4/5$ at every $p\\leq z$\n+by (18), so one constant is uniform along the construction.\n Tracking all constants would be a separate task. Optimizing the bands\n or proving an optimal growth law is also left open. No additional\n logarithmic factor is asserted on the basis of a heuristic multi-cover\n@@ -560,6 +579,11 @@\n   Press, 1974, Theorem 2.2, pp. 68--69 as identified in the project\n   source record. Cited through [KK]. Original printed pages not read\n   in this assignment.\n+- **[FI]** D. Friedlander and H. Iwaniec, *Opera de Cribro*, AMS\n+  Colloquium Publications **57**, 2010, Theorem 6.9 and Corollary\n+  6.10, pp. 68--69. Second carrier for Input S, held at **OCR\n+  custody** (text extraction only); pp. 68--69 have not been read at\n+  page image.\n - **[O]** [OEIS A144311](https://oeis.org/A144311), definition and\n   attribution, revision 23 dated 5 December 2024, inspected\n   18 September 2026. The current entry contains 22 values. They were\n@@ -583,9 +607,14 @@\n   especially sections 3--5. Superseded finite inequality is not reused.\n   SHA-256 `d3de944c59a8c885c599e89b8ba9d3c23c027f0b587f6719c37e1cb14e6373c5`.\n - **[R4]**\n-  [A lower bound for the two-class Jacobsthal function](/projects/twin-primes/docs/paper/kk-lower-bound.md),\n-  live manuscript, sections 2--9 and 11--12.\n-  SHA-256 `7c375d9510a22b6fc2c6241eeffe51c1d92cb9daa17e4eba63220ed1c29c107f`.\n+  [A lower bound for the two-class Jacobsthal function](/projects/twin-primes/history/paper/kk-lower-bound.md),\n+  pinned prior edition of this slug (version 1), sections 2--9 and\n+  11--12. The live path\n+  `/projects/twin-primes/docs/paper/kk-lower-bound.md` now serves the\n+  present revision, so this reference is pinned to the history entry\n+  rather than to the live path.\n+  SHA-256 `7c375d9510a22b6fc2c6241eeffe51c1d92cb9daa17e4eba63220ed1c29c107f`\n+  (version 1).\n - **[R5]**\n   [Search conventions](/projects/twin-primes/docs/research/SEARCH-CONVENTIONS.md),\n   sections 1, 3 and 5.\n","cpu_hours":0,"hashes":{"kk-lower-bound.patch":"d2f5f79e07ffe4d682314cf72fac1df2d85f6e29d3ea08419f6577316e60cafa","paper/kk-lower-bound.md":"15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-25T08:35:46.011Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1538],"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":"kk-lower-bound","revision_path":"paper/kk-lower-bound.md","revision_sha":"15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7","recipe_md":"1. GET <project base>/docs/paper/kk-lower-bound.md -> base sha 8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce (check X-Content-SHA256). 2. GET <project base>/files/15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7 -> revised text, 27396 B. 3. Apply kk-lower-bound.patch (sha256 d2f5f79e07ffe4d682314cf72fac1df2d85f6e29d3ea08419f6577316e60cafa) to the base with patch -p1; output byte-identical to step 2. Prose file, no execution, no CPU.","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":"61cc4c1379045e73daf6b0cff4d1440601038f1f174dfcc567ecd6bad756ff56","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":"2026-09-25T08:35:46.011Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9fb603f4afff75088809f09b","triage_lead":null,"revision_base_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","integration":null,"resolves":[202,203,215],"handle":"Benjaminsen","job_brief":"A reviewer found a defect in the served file `paper/kk-lower-bound.md` while reviewing return #1538 (review #261 by @Benjaminsen), recorded as finding #55. Fix it; do not redo the work it belongs to.\n\nWhat the reviewer said:\n> When #1538's kk-lower-bound.rev2.md is filed on this slug (route 78 next_step), also close three review #73 items that #1538 does not list: (1) Appendix A row 'certified 479,339 at y = 5003 | §10' -> 479,340 (the gap convention, as §10 and Appendix B item 4 now use); (2) §10 'They are not exhibited': drop 'a referee is entitled to call the theorem unfalsifiable ... That is accurate' (review #73 §6: a missing full-scale run is not mathematical unfalsifiability; small structural counterexamples could still refute a proof step); (3) §11.4 rows 'the ledger's O(1)' and 'the exponent assembly': restate the falsifiers (a bounded non-converging residual does not refute O(1); finite agreement does not prove the asymptotic identity), per review #73 §7. Also: the recipe's 'strip one appended newline' step is stale, since the served file hashes to 7c375d95 as fetched.\n\nFetch the current file (GET <project base>/docs/paper/kk-lower-bound.md), make the change, check it still runs and that its stdout reproduces byte for byte elsewhere (progress, timing and rates go to stderr; paths relative to the repository), upload the revised file (POST /files) and return as this job with `\"revision\": { \"path\": \"paper/kk-lower-bound.md\", \"file\": \"<sha256 of the revised file>\" }`, the sha in `files`, a one-line report of what changed and why, and `\"cites\": { \"returns\": [1538] }`. If the file's embedded hashes depend on the change, re-embed them and say so. Send `\"revision\": { …, \"base\": \"<X-Content-SHA256 of the text you edited>\" }` so a later change to the file is caught rather than overwritten, and list the findings your revision answers in `\"resolves\": [<finding ids>]` (GET <project base>/findings?path=paper/kk-lower-bound.md lists the open ones). Accepted, the revision becomes the served version and closes the findings it answered; a finding it leaves open goes to the next fix job.\n\nAlso finding #202 (review #327 of return #1093, @Benjaminsen):\n> [R4] links to paper/kk-lower-bound.md, which this revision (#1093) replaces; its section references (R4 §9 Theorem A, (5.1), §6.3, §11.2) would then resolve to this text. Link R4 to the pinned prior edition (sha 7c375d9510a2…, /history/paper/kk-lower-bound.md) instead of the live path.\n\n\nAlso finding #203 (review #327 of return #1093, @Benjaminsen):\n> Input H (§3, eq. 9): quote Hildebrand–Tenenbaum Cor. 1.3's printed range verbatim (usually cited as y ≥ (log x)^{1+ε}); the stated u ≤ Z^{1−ε} implies it, so the application in §6.3 is unaffected. §9 (lines 492–494): the \"uniformity for these changing residue systems\" item is settled under Input S as quoted, since its constant depends only on κ (HR: κ, A1, A2), which are uniform here (g(p) ≤ 4, g(p)/p ≤ 4/5).\n\n\nAlso finding #215 (review #335 of return #158, @Benjaminsen):\n> §3 Input S: add Friedlander–Iwaniec, Opera de Cribro (AMS Colloq. 57, 2010) Thm 6.9 + Cor 6.10, pp. 68–69, as a second carrier at OCR custody (#158; re-read at OCR in review of #158). Label it OCR, not a page reading, per §10 l.535–537, until pp. 68–69 are read at page image. §8: if the constant is priced, use c < 1/(8C) with C = 2C₂e^{−2γ}·C₁ = 0.41621·C₁, C₁(1/2) = 805.5 (δ = 0.05, K = 3), i.e. c₀ < 3.7·10⁻⁴, conditional on that reading. Do NOT copy #158's 7.5·10⁻⁴: it drops the p = 2 factor.\n","review_deferred":false,"in_triage":true,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1663/transcript","files":[{"sha256":"15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7","name":"paper-kk-lower-bound.md","bytes":27396}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}