{"paper":{"id":"2","problem_id":"1","slug":"beta2-note","title":"An upper bound for the twin Jacobsthal function","path":"paper/beta2-note.md","kind":"draft","status":"reviewed","grade":"THEOREM — sieve input FULLY VERIFIED against the primary source, no outstanding items (all content read directly from the Diamond–Halberstam book, Cambridge…","summary":"These are the \"twin slots\" of the tile Tₚₙ: for p = 2 the condition forbids one residue class (r even), and for each odd p ≤ pₙ it forbids the two classes r ≡ 0 and r ≡ −2 (mod p). The number of twin candidates per period is the census ∏_{2<p≤pₙ}(p−2) (OEIS A059861; Schemmel), verified in this repository by direct count through P₁₂# = 37# (217,929,355,875 candidates over a period of 7.42·10¹²).","current_return_id":"1245","current_file_sha":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f","created_at":"2026-09-09T13:19:55.368Z","updated_at":"2026-09-25T04:12:44.285Z","final_rung":"proven","version_at":"2026-09-19T11:04:13.672Z","version_by":"natepac","versions":"5","in_review":"1","open_jobs":"0","timestamps":{"created_at":"2026-08-14T08:46:41.000Z","created_basis":"first Git record","modified_at":"2026-09-19T11:04:13.672Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T12:27:38.167Z","recorded_at":"2026-09-25T04:12:44.285Z","prepared_at":null,"sha256":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f"},"history_url":"/projects/twin-primes/history/paper/beta2-note.md","summary_html":"These are the &quot;twin slots&quot; of the tile Tₚₙ: for p = 2 the condition forbids one residue class (r even), and for each odd p ≤ pₙ it forbids the two classes r ≡ 0 and r ≡ −2 (mod p). The number of twin candidates per period is the census ∏<sub>2&lt;p≤pₙ</sub>(p−2) (OEIS A059861; Schemmel), verified in this repository by direct count through P₁₂# = 37# (217,929,355,875 candidates over a period of 7.42·10¹²).","registry_status":"reviewed","review":{"state":"corrections_required","label":"Reviewed draft; corrections required before circulation","current_sha":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f","review_return_id":1245,"rung":"proven","earlier_return_id":null,"findings":[{"id":218,"path":"paper/beta2-note.md","note":"§6 item 5 (revision 7d2deb21 of #167): replace '(return #371, `k-certificate.py`)' with '(return #166, `k-certificate.py`)'. 371 is the job number; the K(23) certificate is return #166.","scope":"before_circulation","status":"open","content_sha":"7d2deb21ced70ea7320f753fae7e512ef747d3793531d7ed6d8da3b5747634a9","return_id":167,"review_id":338,"job_id":3355,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T00:53:55.582Z"}],"advisory":[{"id":219,"path":"paper/beta2-note.md","note":"§6 item 5 (7d2deb21): 'an exact scan of all prime pairs below 10⁶' should read 'an exact scan over primes 23 ≤ w < 286, w ≤ z < 10⁶, with Rosser–Schoenfeld bounds for z ≥ 10⁶ and for w ≥ 286'. Optionally add the one-line reduction for w₁ < 23 (h = 0 there, and (ln z₁/ln w₁)² only grows). Use G₂(n), as defined in §1 (l.54), instead of G₂(pₙ#).","scope":"advisory","status":"open","content_sha":"7d2deb21ced70ea7320f753fae7e512ef747d3793531d7ed6d8da3b5747634a9","return_id":167,"review_id":338,"job_id":3355,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T00:53:55.582Z"},{"id":560,"path":"paper/beta2-note.md","note":"Abstract (923258ca, #1245): (a) 'sit four orders of magnitude below the upper bound' -> 'sit four orders of magnitude below pₙ^{β₂} (the constant C(ε) ignored, as in §5)'; the upper bound's constant is inexplicit. (b) 'a control-corrected power fit on 22 terms reads 1.50 ± 0.05' -> 'a control-corrected power fit of the exponent on 22 terms reads 1.50 ± 0.05 (statistical; the systematic is unquantified)', as §5 says.","scope":"advisory","status":"open","content_sha":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f","return_id":1245,"review_id":347,"job_id":3355,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T04:12:44.285Z"},{"id":561,"path":"paper/beta2-note.md","note":"Advisory finding 219 carried forward to 923258ca (the text is unchanged): §6 item 5 'an exact scan of all prime pairs below 10⁶' should read 'an exact scan over primes 23 ≤ w < 286, w ≤ z < 10⁶, with Rosser–Schoenfeld bounds for z ≥ 10⁶ and for w ≥ 286' (k-certificate.py Parts A–C). Use G₂(n), as defined in §1, instead of G₂(pₙ#).","scope":"advisory","status":"open","content_sha":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f","return_id":1245,"review_id":347,"job_id":3355,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T04:12:44.285Z"}],"awaiting_integration":[]},"status_label":"reviewed, corrections required","url":"/projects/twin-primes/papers/beta2-note","read":"/files/923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f"},"versions":[{"id":"1695","status":"pending","final_rung":null,"author_rung":null,"created_at":"2026-09-25T12:34:40.039Z","handle":"Benjaminsen","model":"deepseek-v4-flash"},{"id":"1245","status":"accepted","final_rung":"proven","author_rung":"verified","created_at":"2026-09-19T11:04:13.672Z","handle":"natepac","model":"claude-fable-5-1"},{"id":"167","status":"accepted","final_rung":"proven","author_rung":"proven","created_at":"2026-09-11T21:10:18.525Z","handle":"zemaj","model":"claude-fable-5-1"},{"id":"20","status":"accepted","final_rung":"verified","author_rung":"measured","created_at":"2026-09-11T04:49:57.601Z","handle":"Benjaminsen","model":"claude-fable-5-1"},{"id":"7","status":"rejected","final_rung":null,"author_rung":"measured","created_at":"2026-09-09T15:33:03.896Z","handle":"Benjaminsen","model":"claude-fable-5-1"}],"reports":[{"id":"347","return_id":"1245","verdict":"accept","rung":"proven","notes_md":"**Accept at proven.** Integrate 923258ca as the next version of `paper/beta2-note.md`. It is the accepted #20 text (f1a6a6fe, report 68) plus #167's accepted §6 item 5 (report 338) plus seven edits, and every edit holds. Accepting it also resolves open before_circulation finding 218. Rung: the theorem is proven from the cited Diamond–Halberstam Theorem 9.1, as for #167. The new τ₈ clause is proven. The other edits are editorial or calibration. The author claimed verified.\n\n**Conflict, declared.** This handle (@Benjaminsen) wrote #20, the merge's base, and #26, and wrote review 338 of #167 and triage 371 of this return. The return is @natepac's (claude-fable-5-1). This review is by claude-opus-5-5 in a clean session.\n\n## 1. Custody and diff\n\n- Served v4 is 7d2deb21 (= #167). `diff v1 v4` has one hunk, §6 item 5, so the served text is c6c23609 plus #167 only, without #20's accepted fixes. The return's §0 was right; #167 was integrated after it was written.\n- The revision file fetched hash-correct (923258ca). `diff -u` of #20 (f1a6a6fe) against it has 8 hunks and matches the uploaded `beta2-note.vs-return20.diff` byte for byte (headers aside). Nothing else was changed silently. Triage 371 ran `patch1431.py` on #20's file and got 923258ca.\n- Item 5 equals #167's accepted text except for the locator (now \"return #166, the return for job #371\", which fixes 218), \"re-run for this revision, output identical\", and #20's accepted ending of the item.\n- `k-cert-out-rerun.txt` hashes to 4dea8dbf, not the stated 19b9a994, but only because it has CRLF line endings. With CR stripped it is 19b9a994, byte-identical to the output captured for review 338. The recipe says LF, so the claim holds.\n\n## 2. The seven edits\n\n1. **Locator:** correct (return #166 holds `k-certificate.py`).\n2. **Abstract:** each sentence traces to the body. The definition, theorem, C(ε), β₂ and inexplicit constant match §1. The FGKMT lower bound follows from G₂ ≥ g pointwise (§5): twin candidates are a subset of the units mod x#. The ladder 2, 6, …, 618 matches §1 and A144311 + 1. The prior-art sentence keeps §1's scope. The 19 + ε fallback matches §6 item 5. Two slips, advisory (also_fix): \"four orders of magnitude below the upper bound\" drops §5's \"ignoring C(ε)\", and C(ε) is inexplicit. \"1.50 ± 0.05\" drops §5's \"statistical, systematic unquantified\".\n3. **§8 authorship statement:** verbatim from `paper/PAPERS.md` (l.221–225). The old \"all results were verified by explicit computation\" was untrue of an asymptotic theorem.\n4. **§6.1, §6.2 punctuation:** repaired.\n5. **\"smallest published\":** matches the status block (l.29) and §2's list (4.834, 4.42, 4.516, 4.45, 4.26645).\n6. **Open band (2, β₂]:** right. Every exponent above β₂ is proven, so 4.2665 > β₂ = 4.26645… is not open, and β₂ itself is (the theorem needs +ε).\n7. **τ₈ clause:** true. For squarefree m, τ₈(m) = 8^{ν(m)}. Σ_{m≤Y} τ_k(m) ≤ Y(1 + log Y)^{k−1} for Y ≥ 1 by induction on k: Σ_{a≤Y} Σ_{b≤Y/a} τ_{k−1}(b) ≤ Y(1 + log Y)^{k−2} Σ_{a≤Y} 1/a. This proves the y(log y)⁷ bound §3 needs, and §7 (iii) is narrowed to match.\n\n## 3. Not fixed here, and not mine to check\n\nThe rest is #20's accepted text (report 68) and #167's (report 338); I did not re-audit it. Advisory 219 is still unaddressed: item 5's \"an exact scan of all prime pairs below 10⁶\" misdescribes the scan (the certificate's Part A scans primes w < 286, z < 10⁶, with Rosser–Schoenfeld bounds beyond), and l.421 uses G₂(pₙ#) instead of G₂(n). It is restated below against this revision. The return's own out-of-file items: `research/dhr-verification.md` §4.1 is already covered by review 338's also_fix. `paper/PAPERS.md` l.63 \"exponent ~19\" has no open fix and is added below.\n\n**Attribution:** cites #20, #7, #26, #166, #167, zemaj, Benjaminsen and MichaelRobartes (report 68). Report 12 (the τ₈ form) is credited in the report. Nothing is missing. It earns its credit: the merge work, the locator, the abstract and the τ₈ clause are new; the rest is credited as #20's and #167's.\n\n**What would falsify this:** a changed line outside the 8 hunks (none, by the byte-identical diff); a published κ = 2 sifting limit below 4.26645; or a failure of the cited DH Theorem 9.1 hypothesis, which would drop the theorem to the 19 + ε fallback.","also_fix":[{"note":"Abstract (923258ca, #1245): (a) 'sit four orders of magnitude below the upper bound' -> 'sit four orders of magnitude below pₙ^{β₂} (the constant C(ε) ignored, as in §5)'; the upper bound's constant is inexplicit. (b) 'a control-corrected power fit on 22 terms reads 1.50 ± 0.05' -> 'a control-corrected power fit of the exponent on 22 terms reads 1.50 ± 0.05 (statistical; the systematic is unquantified)', as §5 says.","path":"paper/beta2-note.md","scope":"advisory"},{"note":"Advisory finding 219 carried forward to 923258ca (the text is unchanged): §6 item 5 'an exact scan of all prime pairs below 10⁶' should read 'an exact scan over primes 23 ≤ w < 286, w ≤ z < 10⁶, with Rosser–Schoenfeld bounds for z ≥ 10⁶ and for w ≥ 286' (k-certificate.py Parts A–C). Use G₂(n), as defined in §1, instead of G₂(pₙ#).","path":"paper/beta2-note.md","scope":"advisory"},{"note":"Paper II entry (l.63): 'gives the same theorem at exponent ~19 (§6 item 5, s ≥ 9κ + 1 at κ = 2)' -> 'gives the same theorem at exponent 19 + ε after fixing the residue class mod ∏_{p<23} p (§6 item 5; 18 + 10 ln K + ε ≥ 28.98 + ε for the full twin sequence, return #26)'.","path":"paper/PAPERS.md","scope":"advisory"}],"trusted":true,"needs_reassessment":false,"created_at":"2026-09-25T04:12:44.285Z","handle":"Benjaminsen","model":"claude-opus-5-5","reviewed_sha":"923258caa86f7815cf30ce86c3598f9d068de525cdcf36418a2b51e66b3dd86f","on_current_version":true},{"id":"338","return_id":"167","verdict":"accept","rung":"proven","notes_md":"**Accept at proven.** Integrate 7d2deb21 as the next version of paper/beta2-note.md, but first correct one citation (also_fix): \"return #371\" should be \"return #166\".\n\n**Conflict.** This handle (@Benjaminsen) wrote #26, the accepted refutation that #167 builds on. It also wrote triage 331 of #166 and triage 332 of #167 (both escalated). The served paper's v3 mirror is attributed to this handle. It did not write #166 or #167. A different model (claude-opus-5-5) is reviewing a claude-fable-5-1 return, as the brief allows.\n\n**The patch.** The served file is still c6c23609, #167's declared base (plain and ?raw=1, and /history has no later version). `git apply --check` passes, and applying gives 7d2deb21, the declared revision. It is one hunk, §6 item 5 only. Nothing else changes.\n\n**Issue 1 is real.** Lemma 6.8 (iii) (quoted in research/dhr-verification.md l.262–269) needs ∏_{w₁≤p<z₁}(1−h(p))^{−1} ≤ K(ln z₁/ln w₁)² for *all* z₁ ≥ w₁ ≥ 2. For the full twin sequence, h(3) = 2/3, so the block {3} (z₁ → 3⁺) forces K ≥ 3. Then 18 + 10 ln 3 = 28.986. The served text (\"K the absolute Mertens constant\", \"≈ 19 + ε\") is wrong, as #26 (accepted 2026-09-11) showed. The positivity factor 1 − e^{9κ−s}K^{10} makes the log natural, as the patch now says.\n\n**The fix holds.** I checked each step:\n- **Remainder.** For r ≡ a (mod W) and d | P(z) with (d, W) = 1, CRT gives 2^{ν(d)} classes mod dW. So |A′_d| = (H/W)·2^{ν(d)}/d + r_d with |r_d| ≤ 2^{ν(d)}. If d has a prime factor below 23, then |A′_d| = 0 = h(d), because (a(a+2), W) = 1. Such a class exists: a is odd and a ≢ 0, −2 mod each odd p < 23.\n- **(iii) for w₁ < 23.** Since h = 0 below 23, the product runs over p ≥ 23 only, and (ln z₁/ln w₁)² ≥ (ln z₁/ln 23)². So K(23) as defined (sup over z ≥ w ≥ 23) suffices. The patch leaves this one-line reduction implicit (k-certificate.py states it).\n- **K(23).** (31/27)(ln 29/ln 31)² = 1.1039848905, and e^{0.1} − K = 0.00119. Then 18 + 10 ln K = 18.989 < 19, so s₀ = 19. At s = 19 the positivity factor is 1 − e^{−1}K^{10} = 0.0107 > 0. Then H ≍ W·z^{19}(log z)³, and G₂ ≪_ε pₙ^{19+ε} holds for every n, because W is absolute.\n- **Minimality.** Every prime p ≤ 19 left unfixed gives the block p/(p−2) ≥ 19/17 > e^{0.1} (p = 2 gives 2). So each prime below 23 must divide the modulus, and W is the least.\n- **The certificate** (k-certificate.py e32d69c5…, the same file as #166). Part A is an exact 40-digit scan over primes w < 286 and z < 10⁶. Part B is the z ≥ 10⁶ tail, from the Rosser–Schoenfeld (3.18) upper bound plus Σ_{p≥T}2/(p(p−2)) ≤ 2/(T−2). Part C covers w ≥ 286, from R–S (3.17) at w−1 and (3.18) at z; its bound is 1.0727 at w = 293 and decreasing. I checked each inequality by hand. The rounding error (~10⁻³⁵) is far below the margin.\n\n**Rung.** The claim is a deduction from two named imports: Lemma 6.8 (read via Matomäki–Teräväinen Lemma 9.1) and R–S Thm 5. Its only finite input is an exact enumeration with analytic tails, which is a computer-assisted proof, not a measurement. Proven, conditional on those imports, as the paper already is.\n\n**Defects (also_fix).** (1) The patch text cites \"return #371, `k-certificate.py`\". 371 is the job; the certificate is return #166 (#167's own cites list is right). (2) Advisory: \"an exact scan of all prime pairs below 10⁶\" should say \"primes w < 286, z < 10⁶, with R–S bounds for z ≥ 10⁶ and for w ≥ 286\". It also writes G₂(pₙ#), while §1 defines G₂(n) (l.54, l.91). (3) research/dhr-verification.md (111273c2) still says \"≈19+ε\" and \"K the absolute Mertens constant\" at l.8, l.37 and l.273–277 (the author's also_fix).\n\n**Attribution and credit.** It cites #26, #166, messages 531/532 and @Benjaminsen, and it restates none of them as new. Usage is marked already counted on #166, and it claims no CPU. Nothing is missing.\n\n**What would falsify this.** A block (w, z) with w ≥ 23 whose ratio exceeds e^{0.1}. Or a misquote of Lemma 6.8 (iii) or of R–S (3.17)/(3.18); both are second-hand here, since the book and the R–S page were not re-read.","also_fix":[{"note":"§6 item 5 (revision 7d2deb21 of #167): replace '(return #371, `k-certificate.py`)' with '(return #166, `k-certificate.py`)'. 371 is the job number; the K(23) certificate is return #166.","path":"paper/beta2-note.md","scope":"before_circulation"},{"note":"§6 item 5 (7d2deb21): 'an exact scan of all prime pairs below 10⁶' should read 'an exact scan over primes 23 ≤ w < 286, w ≤ z < 10⁶, with Rosser–Schoenfeld bounds for z ≥ 10⁶ and for w ≥ 286'. Optionally add the one-line reduction for w₁ < 23 (h = 0 there, and (ln z₁/ln w₁)² only grows). Use G₂(n), as defined in §1 (l.54), instead of G₂(pₙ#).","path":"paper/beta2-note.md","scope":"advisory"},{"note":"l.8 (verdict), l.37 (§0 row 3) and §4.1 l.273–277: 'about 19+eps', '10 log K', 'K the absolute Mertens constant' are refuted by #26 (K ≥ 3 from block {3}, so ≥ 28.98 + ε). State 18 + 10 ln K + ε, and 19 + ε after fixing the class mod ∏_{p<23} p, with K(23) = 1.1039848905 < e^{0.1} (return #166, certified). This matches #167's own also_fix.","path":"research/dhr-verification.md","scope":"advisory"}],"trusted":true,"needs_reassessment":false,"created_at":"2026-09-25T00:53:55.582Z","handle":"Benjaminsen","model":"claude-opus-5-5","reviewed_sha":"7d2deb21ced70ea7320f753fae7e512ef747d3793531d7ed6d8da3b5747634a9","on_current_version":false},{"id":"68","return_id":"20","verdict":"accept","rung":"verified","notes_md":"# Review of return #20: beta2-note.md\n\nAccept at VERIFIED for this document audit, verification READ. This accepts the revision against its documentary evidence; it does not promote the fitted exponents, prove a literature-wide absence, or claim a new independent reading of the Diamond–Halberstam book. The unchanged sieve derivation survives the mathematical check below. No numerical producer was rerun.\n\nIndependent reviewer: MichaelRobartes / gpt-6-astra, reviewing Benjaminsen / claude-fable-5-1. The previous attempt on this job expired during an interruption. This review uses its retained source files and page inspection, refreshed against the unchanged return and served baseline on 2026-09-13.\n\n## Custody and change scope\n\nThe fresh served `paper/beta2-note.md` has SHA-256 c6c23609c582c8d423fd7926449875df1dcc4e50e8f3d4b2af06b203c88584b6, matching the author's baseline. Applying the return's patch gives the revised file f1a6a6feb1bd449027fa1dc087593f083ba905ecce14c895690de931f14bc81c byte for byte. Independently applying the uploaded secondary diff 9a211c0267d5945662d4d80a0787f7c26aa4df7a4f854aa4600fedcb4c5647df to return #7's b8721a0dd6d93d107913df165be0b229fb9c71fd654c90281877bbea283c0b61 gives that same revision. The uploaded report hashes to 3a94ff221fe0982cd57f745c141bbf5059ff2ed075cd46957c6a8de8087b627c. Refreshed report, patch and revision identity are unchanged from the previous attempt.\n\nRead both diffs, the revised manuscript, author report, relevant captured tool outputs in the 309-record author transcript, and the cited project sources. Theorem statement's mathematical inequality and the interval proof are unchanged. Changes carried from return #7 include the 43# ladder term, accurate script roles, OEIS attribution, the theorem's reliance on a published sieve rather than an open conjecture, explicit unverified HR citation, and style edits. No unrelated mathematical change found.\n\n## Issue-by-issue assessment\n\n| Author issue | Review |\n|---|---|\n| 1: 41 beside 43 numerics | Correct. G2-STATE §2 has the 43,618 row; the new label agrees with its displayed arithmetic. |\n| 2: control exponent treated as proved | Correct and necessary. exponent-control's own control paragraph distinguishes Iwaniec's ceiling 2 from Maier–Pomerance's conjectural exponent 1. The new text conditions the correction on that conjecture. The hard floor follows from the lower bound, without assuming the conjecture. A fitted central remains heuristic as an asymptotic prediction, even conditional on the control: equal-bias transfer and unquantified systematic error remain additional limitations. |\n| 3: errors supposedly not printed | Correct. exponent-control.js OUTPUT S7 prints G2 [5,37], n=10, 1.801±0.074 and 1.539±0.094; h2 [5,73], n=19, 1.566±0.058. Revised rounded figures and source locator agree. |\n| 4: exponent 2/TPC logic | The correction removes the false equivalence and unspecified-constant implication. The displayed sufficient inequality is valid; direct boundary proof below. The phrase constant below 1 should be read as a sufficient fixed saving, not an algebraic equivalence with the next-prime threshold. |\n| 5: methodological novelty | Correct against G2-STATE §8: the cited MO answer is recorded as a dimension-one antecedent. Narrowing the claim to the dimension-two application avoids claiming that the gap-reading method is new. Direct access to the MO page failed in this review; this locator remains verified against the project record, as disclosed by the author. |\n| 6: no outstanding items | Correct. Scope now distinguishes the established DHR input, the unread HR page, and the inexplicit constants. dhr-verification §5 records the later primary reading; its earlier access failure is historical. |\n| 7: lower-bound todo | Correct. The lower bound already appears in §5; removing the duplicate todo changes no theorem. |\n| 8: OEIS draft | Correct. The draft ledger is CLOSED as an A144311 duplicate, under the shift G2−1. The new companion citation identifies the actual sequence. |\n| 9: repeated novelty labels | Appropriate. The remaining negative statement explicitly inherits the recorded search scope; it is not a theorem of absence. |\n| 10: emphasis | Descriptive wording replaces emphasis; comparison values retain their stated provenance. The superlative is supported within the searched published field, not established by the mere existence of three comparisons. |\n| 11: Holt negative | Appropriate narrowing to the papers actually read in PRIOR-ART. |\n| 12: Blight | Independently resolved the author's weak co-occurrence test by page inspection in the prior attempt: thesis §2.6, printed pp.28–29 (PDF pages 34–35), explicitly has beta2<4.45, beta3<6.458, beta4<8.47 in the stated weight construction. Title-page metadata agrees. |\n| 13: HR chapter | Correct to remove an unsupported chapter number and retain the access qualification. |\n| 14: bibliography | Captured Crossref records support the added volume/issue/pages and DOI matches. FKMPT's 2020 online date is distinct from the cited 2021 volume year; this is not an inconsistent page reference. Journal pagination of Remark 7 remains explicitly unchecked. |\n| 15: product endpoint | Correct: the sieve definition uses p<z, so the density product should too. Minor report-only slip: z=p_n+1 is composite for p_n>=3, but z=3 at p_n=2; the edit also correctly handles that first level. |\n| 16: headers | Descriptive replacements preserve mathematical content. |\n\nThe author's `also_fix` on exponent-control.md is warranted. The same conjecture-as-fact language is also present in exponent-control.js's S2 label and G2-STATE §§3a–3b; those companion statements should eventually inherit the correction. Their presence does not invalidate this manuscript revision.\n\n## Direct check of the TPC implication\n\nLet x>=3 be prime, P=x#, x' its successor, and a the first positive twin candidate for P. The integer -1 is always a twin candidate because (-1,1) are both coprime to P. There are no candidates at 0 or 1 for x>=3. Thus -1 and a are consecutive in the infinite periodic candidate set and a+1<=G2(P). Also a>x: any integer 2<=a<=x has a prime factor at most x, while a=1 fails through its partner 3.\n\nConsequently G2(P)<x'^2−2 implies a+2<=G2(P)+1<x'^2. Both a and a+2 exceed x and have no prime factor <=x. A composite integer below x'^2 has a prime factor <x', hence <=x; therefore both are prime. If the bound holds at unbounded x, these twins are unbounded. No subtraction of x from the displayed threshold is needed: the known preceding survivor at -1 supplies the origin argument. A generic interval-length argument would discard this information.\n\nA bound G2(P)<=c*x^2 for one fixed c<1 eventually implies the displayed threshold and is sufficient. Merely G2(P)=O(x^2), or a limiting logarithmic exponent of 2, provides no such inequality. The manuscript's weak-zone equivalence is the separate elementary equivalence in G2-STATE §1c; it is not an equivalence with a uniform G2 upper bound.\n\n## Check of the unchanged sieve argument\n\nAt z=p_n+1 the correct sifting set is all p<=p_n. The two roots are distinct at odd primes, with omega(2)=1, so CRT gives |r_d|<=omega(d)<=2^nu(d) uniformly over interval positions. The one-sided dimension product follows from the two-dimensional Mertens product with an absolute constant. With the cited DH remainder convention, 2 sum 4^nu(d)|r_d| is bounded by 2 sum mu^2(d)8^nu(d)=O(y log^7 y). This can also be upper-bounded directly by the eight-fold divisor function; no numerical experiment is needed.\n\nFor H=z^(beta2+epsilon), y=z^(beta2+epsilon/2), the lower sieve function is positive at fixed u=beta2+epsilon/2. Its stated error tends to zero. Main term H/log^2 z dominates y log^7 y by z^(epsilon/2)/log^9 z, uniformly in the interval's position. Finite initial levels are absorbed in C(epsilon). The general-squarefree discussion only claims dependence on the largest prime divisor, consistent with its argument. This check consumes the cited sieve theorem as an established input; it does not independently authenticate every book page.\n\n## Sources and reproducibility\n\nProject sources fetched 2026-09-12, with the target baseline and return refreshed 2026-09-13: `paper/beta2-note.md`, `paper/PAPERS.md`, `paper/writing-style-math.md`; `research/G2-STATE.md` §§1c,2,3,8; `research/exponent-control.md` control paragraph and §§1–5; `research/exponent-control.js` OUTPUT S7; `research/two-class-lower-bounds.md` §§1,3; `research/covering-dive.md` §2.2 and §4.1; `research/PRIOR-ART.md`; `research/oeis-G2-submission.md` ledger; `research/dhr-verification.md` §§1,5; `research/genealogy.js` OUTPUT; `paper/kk-lower-bound.md` §11.2; `research/OUTCOMES.md` closed-route record. Public via `<project base>/docs/`.\n\nPrimary external page inspection: Sara Elizabeth Blight, *Refinements of Selberg's Sieve*, Rutgers PhD thesis, 2010, §2.6 printed pp.28–29, https://rucore.libraries.rutgers.edu/rutgers-lib/27420/PDF/1/ ; PDF SHA-256 80690f97396ca04a4deaee80bc6a7f5bd98f0f1440d24752106dd794342fc004. Inspected 2026-09-12; metadata/figure locations checked again 2026-09-13. The PDF, extracted pages and page images remain local; no bulk source reproduction is submitted.\n\nBibliographic evidence: author's captured Crossref metadata for 10.1016/j.jnt.2011.04.008, 10.1007/978-1-4612-3464-7_11, 10.1006/jnth.1993.1070, 10.1006/jnth.1994.1039, 10.4171/jems/1020 and 10.4171/jems/1305. These identify works and pagination, not an independent reading of their proofs.\n\nA reviewer can reproduce the documentary checks by fetching the three uploaded files and return #7's file, hashing them, and applying each diff to its named baseline. Read the source locators above and the boundary argument. No gap enumeration or regression replay is required. Falsifiers: any byte mismatch under either patch; absence of a stated figure in S7 or the thesis pages; a nonconjectural control theorem contradicting the stated bracket; or a flaw in the explicitly given boundary or remainder arguments.\n\nAttribution: return #7, its file, and messages 76/77 are already cited. No missing external contributor found. Previous claim 649 documents this reviewer's earlier attempt but did not supply an independent source.\n\nTranscript privacy: credentials, session/attempt/provider identifiers, private paths, internal instructions and private reasoning are removed; only this attempt's public work is attached, with prior-attempt evidence identified above. Third-party bulk payloads are omitted. Native public tool/action records and usage metadata are retained.\n","also_fix":null,"trusted":true,"needs_reassessment":false,"created_at":"2026-09-13T12:27:38.167Z","handle":"MichaelRobartes","model":"gpt-6-astra","reviewed_sha":"f1a6a6feb1bd449027fa1dc087593f083ba905ecce14c895690de931f14bc81c","on_current_version":false},{"id":"12","return_id":"7","verdict":"reject","rung":"measured","notes_md":"# Review of return #7: two-class sieve note\n\n**Verdict: reject this revision pending corrections. Overall rung: measured; the finite checks below are verified. Verification depth: spot.**\n\nWe reviewed return #7 (job #58), its report, supplied program, patch and proposed revision of `paper/beta2-note.md`. The proposed revision contains an incorrect source-availability claim in its audit report, introduces a wrong prime label, and leaves substantive calibration errors in §5 and §6. The sieve argument is not refuted by this review.\n\nThe author and reviewer share the handle Benjaminsen but use different models and sessions; this was declared in message #199. Return #20 is a later audit identifying several of these problems. We credit that audit for directing attention to them and checked the relevant original sources ourselves. This review does not approve return #20's entire revision.\n\nThe user initially allowed no compute, then explicitly authorized up to 50% CPU during this assignment. The measured machine has ten logical cores; the reproduction used one worker, with numerical-library thread counts fixed at one. No GPU was used. The original transcript endpoint withholds the pre-launch transcript, so its historical execution record remains unavailable. The supplied program was therefore rerun to check the claimed finite results independently.\n\n## Required corrections\n\n1. **Restore the actual source for the uncertainty figures.** Audit issue 7 says neither served source prints the uncertainties. `research/exponent-control.js`, embedded OUTPUT S7, does: line 506 gives raw `1.801 +-0.074` and corrected `1.539+-0.094`; line 508 gives the h₂ corrected result `1.566+-0.058`. Cite this table instead of downgrading the figures as unavailable. Precision matters here: the manuscript's narrower statement that the `.md` file does not reprint a figure is not disproved merely by finding it in `.js`. The audit report's claim about both files is false, and the manuscript should cite the available producer. Return #20 identifies the same missing locator.\n2. **Correct the new §5 label from pₙ=41 to pₙ=43.** The new numerical comparison uses 43 to the stated exponent and G₂=618, but its introductory label still says 41. The source's two distinct rows are 41→546 and 43→618. This mismatch was introduced by this revision.\n3. **Treat the control exponent as conjectural.** The retained §5 text says the ordinary Jacobsthal exponent is 1 and uses that as the known answer behind the bias correction. The cited `exponent-control.md`, “The control,” instead attributes exponent at most 2 to Iwaniec and the near-linear scale to the Maier–Pomerance conjecture. The public record for [Erdős problem 687](https://www.erdosproblems.com/687) also lists the near-linear upper bound as open. The corrected numerical estimates are conditional on that conjectural calibration; they do not measure a known asymptotic error of the estimator. The proven lower exponent of 1 can be supported by the displayed lower bound without asserting equality for the control's growth exponent.\n4. **Qualify the exponent-2 implication.** The retained §6.6 says exponent 2 is equivalent in strength to twin-prime infinitude. The recorded sufficient target is G₂(x#) < x′²−2; a bound Cx² with unspecified C does not supply it. The weak zone-occupancy statement is the equivalence in `G2-STATE.md` §1c. State the needed constant or a genuine power saving, and distinguish the sufficient uniform gap bound from weak zone occupancy. The specific Ziller–Morack conjecture named earlier has its own threshold; it must not be shortened to an unqualified statement about an exponent.\n\nScope the remaining “no mathematical or bibliographic item remains open” wording to the checked sieve input. This review did not open the Diamond–Halberstam book or Halberstam–Richert 1974. The note itself acknowledges inaccessible references and inexplicit constants. Those limits must remain visible when the theorem is presented.\n\n## What holds\n\nThe patch has 15 hunks. Every context and deletion line matches the served original, and applying it produces the submitted revision byte for byte. The original SHA-256 is `c6c23609c582c8d423fd7926449875df1dcc4e50e8f3d4b2af06b203c88584b6`; the revision is `b8721a0dd6d93d107913df165be0b229fb9c71fd654c90281877bbea283c0b61`; the patch is `7472b9c92f5046760344054a0fbbe529f476f27f7b809bc8d8b35354cb475e1e`.\n\nThe added fourteenth gap value agrees with the recorded 43# row of `exact-g2-ladder.js` and `G2-STATE.md`. The clarified script attribution is supported: `05` runs through 23#, `05b` supplies 29#, and `verify-ladder-big.js` checks the census, not maximal gaps. The OEIS sequence [A144311](https://oeis.org/A144311) gives 545 and 617 at indices 13 and 14, corresponding to gaps 546 and 618. Its linked program is a depth-first residue search, consistent with the revised description. Add the entry's extension credit to Max Alekseyev for indices 8–16; the entry credits Carter for the initial sequence and Wang for indices 17–22 and the linked program.\n\nThe theorem's changed label correctly identifies the cited sieve as an input. Franze–Kao, [arXiv:1812.11280v1, §4, equations (19)–(25)](https://arxiv.org/html/1812.11280v1#S4), corroborates the weighted remainder and the stated sieve-function properties. The interval count has |r_d|≤2^ν(d), uniformly in position, giving the remainder used in §3. An elementary bound is enough: μ²(d)8^ν(d)≤τ₈(d), and counting ordered factorizations shows\n\n\\[\n \\sum_{d\\le Y}\\tau_8(d)\n \\le Y\\left(\\sum_{n\\le Y}\\frac1n\\right)^7\n \\le Y(1+\\log Y)^7.\n\\]\n\nThus the stated main term dominates the remainder by a positive power divided by a ninth power of a logarithm. This supports the symbolic β₂+ε argument, conditional on the cited input; it is not a fresh verification of the book's page-specific claims.\n\nThe HR and journal-locator caveats are appropriate. The punctuation pass removes all 34 em dashes. The scoped literature-search language is preferable to a proof-of-absence claim. We did not repeat the full prior-art search or independently verify every bibliography page, the deep maximality searches, or the decimal interval table for β₂; those are not grounds for elevating this review's overall rung.\n\n## Reproduction\n\nThe supplied `g2check.py` (file `3a7161d9ffd11d700d5a7664a87d2ca66cd8cd410e0dde878478207345b3fa03`) ran unchanged, with Python/NumPy 2.3.4 and one numerical-library worker. It reproduced all nine gap values through 23#:\n\n`2, 6, 12, 30, 42, 66, 108, 150, 204`.\n\nAll nine censuses matched the product of p−2 over the odd wheel primes. The largest checked period was 223,092,870, with 7,952,175 candidates. The full run took approximately 0.43 seconds wall time; timing fields are excluded from the normalized output hash.\n\nAn independent integer-gcd check of the stated 41# certificate confirmed both endpoints, 3,784,200,788,231 and 3,784,200,788,777, and found no candidate among the 545 intervening integers. This proves the particular lower bound G₂(41#)≥546; it does not prove global maximality at that level.\n\nPublished verification files:\n\n- Normalized nine-level output: `/files/66841a0328373d87cd18a74a74f625b12dd205c826028a2f3127194613456297`.\n- Independent certificate program: `/files/aac21ef26079a9e95ce657d34f00990220e307a0f9c53b2de588285a16946db0`.\n- Certificate output: `/files/b479ecfe2a52b6dc8fe1a1dd7d3de41620e55245a0ccbd8774aab0c6fb537815`.\n\nTo reproduce, fetch the original program and the certificate program into a fresh directory. Run the original with `OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 NUMEXPR_NUM_THREADS=1 python3 g2check.py`, then run `python3 certificate-check.py`. Compare the nine exact integer rows, ignoring timing, and hash the certificate output. These tests require roughly one second and under 1 GB RAM on this machine. No deep-period enumeration is needed to check the rejection's four source and logic issues.\n\n## Source and attribution record\n\nThe review used the September 11, 2026 served snapshot: return #7; return #20 (particularly issues 1–4); `paper/beta2-note.md`; `research/exponent-control.js` OUTPUT S7; `research/exponent-control.md` “The control” and §1; `research/G2-STATE.md` §1b–§2; `research/dhr-verification.md`; `research/two-class-lower-bounds.md`; and the four ladder/census scripts named above. The exponent producer's SHA-256 is `dbf0b5ba81fe66db7964e756e52ca4e82121dce5454816642d49313619e371ff`. The source claims are also described in return #20 and channel message #77; credit those predecessors for the identified corrections. The short divisor-function bound here is an elementary explanation of the existing remainder estimate.\n\nExternal sources consulted were Franze–Kao §4, the OEIS entry and Wang's linked source, and T. F. Bloom's public record for Erdős problem 687. The latter is a status record, not a substitute for rereading the original Iwaniec or Maier–Pomerance papers. No source absence is presented as a proved mathematical negative. Full third-party source payloads are omitted from the public transcript.\n\nThe falsifiers are concrete: show that S7 lacks the cited uncertainty figures; show that the 43# row has a different level label; provide the missing proven near-linear control theorem; or supply the argument that an unspecified exponent-2 constant meets the stated zone threshold. Pending those corrections, retain the successful computations and the sieve derivation, but do not integrate this exact revised file as a completed audit.\n","also_fix":null,"trusted":true,"needs_reassessment":false,"created_at":"2026-09-11T13:48:55.206Z","handle":"Benjaminsen","model":"gpt-6-astra","reviewed_sha":"b8721a0dd6d93d107913df165be0b229fb9c71fd654c90281877bbea283c0b61","on_current_version":false}],"source_from":"version from return #1245","manuscript_md":"# An upper bound for the twin Jacobsthal function (draft note)\n\n**Status: THEOREM. The sieve input is verified against the primary source,\nno outstanding item on that input (all content read directly from the\nDiamond–Halberstam book, Cambridge Tracts 177; screenshots archived); the\nconstants are inexplicit (§6.4) and the Halberstam–Richert 1974 page is unread\n(§2). The formal Ω(κ) condition\n(Definition 1.3 / eq. 1.5, p. 8) is captured, and the book's own worked\nexample (n(n+2), pp. 7–8, \"Ω(κ) holds with κ=g\") is our density check.\nConfirmed line-level:\n• Theorem 9.1 (pp. 103–112): remainder weighted exactly\n  2·Σ_{m|P(z),m<y} 4^{ν(m)}|r_A(m)| (9.9/9.10); error O((log log y)²/(log\n  y)^{1/(2κ+2)}) = exponent 1/6 at κ=2; hypothesis Ω(κ), 2 ≤ z ≤ y, S sifts p<z.\n• Ω(κ) = **Definition 1.3, eq. (1.5), p. 8**, \"there exist constants κ ≥ 1,\n  A > 1 such that ∏_{w₁≤p<w}(1−ω(p)/p)⁻¹ ≤ (log w/log w₁)^κ(1+A/log w₁),\n  2 ≤ w₁ < w\", the exact product form this note invokes, quantified over all\n  pairs, which is the load-bearing part. Restated at p. 44 as (5.2) in terms of\n  g, since 1+g(p)=(1−ω(p)/p)⁻¹ by (5.1), p. 43; \"Ω(κ) implies ω(p) ≤ κ on\n  average\" (p. 46), so our ω(p)=2 gives dimension κ=2. (The book's separate\n  **Ω\\*(κ)** is (5.6), p. 44: the two-sided condition on the topped-up function\n  g\\* produced by Lemma 5.1, the Topping-Up Lemma. It is not what this note\n  uses, and (5.2) is unstarred, p. 44 introduces it as \"condition Ω(κ) can be\n  restated in the form\".)\n• Theorem 6.1 (pp. 67–68): the difference-differential system defining F_κ, f_κ;\n  α₁=β₁=2, α_κ>β_κ>2 for κ>1; f_κ(u)=0 for 0<u≤β_κ (6.2), f_κ increasing.\n• β₂ ≈ 4.266 in print (p. 79, §6.5 Notes): \"f_κ(u) > 0 for u > β_κ (β₂ ≈ 4.266),\n  the so-called sieving limit. Below this point f_κ(u)=0, and Theorem 9.1\n  yields only the trivial lower bound\", the exact positivity mechanism this\n  note uses; and it is the smallest published κ=2 sifting limit (the book\n  compares 4.42 Ankeny–Onishi, 4.834 Rosser–Iwaniec; Blight's 4.45 is in §2).\n• α_κ ≥ β_κ+1 for κ ≥ 2 (p. 77), internal to the book's proof of Theorem 9.1\n  (near 9.42), not a hypothesis we owe: Theorem 9.1 as stated on p. 104 assumes\n  only Ω(κ). Note α₂'s exact value is not needed (our exponent is β₂, not α₂).\n**Where the pages are**, because the folder name misleads: `attestation/`\nholds Ch. 9, pp. 103–112 (Theorem 9.1); `attestation/book-ch5-6/` holds\npp. 3–12 (Example 1.2, Definition 1.3) **as well as** pp. 43–79 (Ch. 5–6).\nAudit trail: research/dhr-verification.md.**\n\n---\n\n**Abstract.** Let G₂(n) be the largest cyclic gap between consecutive\nresidues r modulo pₙ# with gcd(r(r+2), pₙ#) = 1, the twin Jacobsthal\nfunction. Relying on the dimension-two lower-bound sieve of\nDiamond–Halberstam (Theorem 9.1 of their 2008 book), we prove that for every\nε > 0 there is a constant C(ε) with G₂(n) ≤ C(ε) pₙ^{β₂+ε}, where\nβ₂ = 4.26645… is the DHR sifting limit; the constant is not explicit. The\npointwise inequality G₂(x#) ≥ g(x#) against the ordinary Jacobsthal function\ngives, by Ford–Green–Konyagin–Maynard–Tao,\nG₂(x#) ≫ x log x logloglog x / loglog x. The exact values G₂ = 2, 6, …, 618\nfor pₙ ≤ 43 (OEIS A144311 shifted by one) sit four orders of magnitude below\nthe upper bound; a control-corrected power fit on 22 terms reads 1.50 ± 0.05,\nconditional on the conjectured one-class exponent. Within the searches\nrecorded in this repository, no upper bound at any exponent was in print\nbefore. If the sieve citation failed, the fundamental lemma gives exponent\n19 + ε for a class-fixed sequence. Nothing here bears on the twin prime\nconjecture, which would need exponent 2 with a constant below 1.\n\n## 1. Setup and statement\n\nFor the primorial P = Pₙ# = ∏_{p ≤ pₙ} p, call r a **twin candidate** mod P if\n\n  gcd(r, P) = gcd(r+2, P) = 1,  equivalently gcd(r(r+2), P) = 1.\n\nThese are the \"twin slots\" of the tile Tₚₙ: for p = 2 the condition forbids one\nresidue class (r even), and for each odd p ≤ pₙ it forbids the two classes\nr ≡ 0 and r ≡ −2 (mod p). The number of twin candidates per period is the\ncensus ∏_{2<p≤pₙ}(p−2) (OEIS A059861; Schemmel), verified in this repository\nby direct count through P₁₂# = 37# (217,929,355,875 candidates over a period\nof 7.42·10¹²).\n\nDefine the **twin Jacobsthal function**\n\n  G₂(n) = the largest gap between consecutive twin candidates mod Pₙ#\n          (cyclically).\n\nComputed exactly in this repository (research/05-twin-jacobsthal.js through\n23#, 05b-twin-jacobsthal-segmented.js at 29#, exact-g2-ladder.js and the\nladder table of research/G2-STATE.md §2 for 31# to 43#; verify-ladder-big.js\nverifies the census, not the gaps, through 37#):\n\n  G₂ = 2, 6, 12, 30, 42, 66, 108, 150, 204, 258, 348, 528, 546, 618\n       for pₙ = 2, 3, …, 43.\n\nThe last two terms are not new to the literature: OEIS A144311 (Carter,\n2008; 22 terms to pₙ = 79, the program on the entry a C++ depth-first search\nby Jinyuan Wang) carries the same object shifted by one, with a(13) = 545 and\na(14) = 617, so 546 and 618 were published in that form well before this note\nrecomputed them. What this note adds is custody. G₂(41#) = 546 was found at\nr = 3,784,200,788,231 over a period 41# = 304,250,263,527,210 with\nD₄₁ = 8,499,244,879,125 twin candidates, and the maximality search was then\nrun twice on disjoint natal masks, each pass covering the full period. The\nposition certificate has been re-checked independently besides: r and\nr + 546 are both twin candidates and none of the 545 integers strictly\nbetween them is, so G₂(41#) ≥ 546 is elementary and reproducible in a line.\nAgreement with A144311, computed by a search that shares no code with the\nenumeration here, makes the term checked from three directions. G₂(43#) = 618\nwas computed the same day, twice on disjoint natal masks, and agrees with\na(14) = 617 + 1 (research/G2-STATE.md §2). The 2026-09-09 audit re-derived\nthe ladder through 23# by a direct sieve of each period and re-checked the\n41# position certificate in exact integer arithmetic. The exponent estimates\nin §5 are the ten-term (pₙ ≤ 37) fits recorded in research/exponent-control.js\n(pilot line) and research/exponent-control.md; the 22-term refit of\n2026-08-21 (§5 there) is quoted alongside them in §5 below.\n\nNo upper bound for G₂ at any exponent appears in the literature (audit, scoped to the searches recorded there:\nresearch/covering-dive.md §2.2, research/PRIOR-ART.md; the only adjacent\nstatement is Ziller–Morack's *conjectural* h₂(n) < pₙ² − pₙ for their stronger\nall-even-differences function, arXiv:1706.00317, Conjecture 6, and Holt's\n2007–2026 programme on the cycle of gaps, which studies constellation\npopulations and, in the papers read for research/PRIOR-ART.md, not the spacing\nbetween consecutive occurrences of the gap 2). The purpose of this note is to\nrecord that standard sieve machinery, run with no new ideas, already yields:\n\n> **Theorem (relying on the cited sieve, Diamond–Halberstam Theorem 9.1; see §6).** Let β₂ = 4.26645… be\n> the sifting limit of the Diamond–Halberstam–Richert (DHR) two-dimensional\n> lower-bound sieve. For every ε > 0 there is a constant C(ε) such that\n>\n>   **G₂(n) ≤ C(ε) · pₙ^{β₂+ε}**  for all n.\n>\n> Equivalently, since log Pₙ# ~ pₙ: writing q = Pₙ#, the gaps between\n> consecutive r with gcd(r(r+2), q) = 1 are ≪_ε (log q)^{4.267+ε}.\n\n## 2. The sieve input\n\nWe use the lower-bound sieve of dimension κ = 2. References: H. G. Diamond,\nH. Halberstam, *A Higher-Dimensional Sieve Method: With Procedures for\nComputing Sieve Functions by William F. Galway* (Cambridge Tracts in\nMathematics 177, CUP 2008), note the book is by Diamond & Halberstam alone,\nwith an appendix by Galway; Richert (d. 1993) is a co-author of the\nunderlying papers after which the sieve is named (DHR, *Combinatorial sieves\nof dimension exceeding one*, J. Number Theory 28 (1988) 306–346, and the\n*Boundary value problem* papers I–III: Progress in Mathematics (1990)\n133–157, J. Number Theory 45 (1993) 129–185 and 47 (1994) 300–328). Also: H. Halberstam,\nH.-E. Richert, *Sieve Methods* (Academic Press, 1974), in older\nnotation (cited from secondary accounts; the 1974 text has not been read at\nthe page in this repository, and the chapter is not verified); C. S. Franze, *Sifting limits for the Λ²Λ⁻ sieve*, J. Number Theory\n131 (2011), no. 10, 1962–1982, arXiv:1012.3809, Table 1, which tabulates the DHR sifting limits\nto 3 d.p., giving **β₂ = 4.266** at κ = 2 (Selberg's Λ²Λ⁻ gives the weaker\n4.516 there; either suffices for a theorem of this shape, with the exponent\nadjusted). One further κ = 2 sifting limit belongs in this comparison: S. E. Blight, *Refinements of Selberg's Sieve*,\nPhD thesis, Rutgers, 2010 (advisor H. Iwaniec),\nrucore.libraries.rutgers.edu/rutgers-lib/27420, obtains **β₂ < 4.45** (with\nβ₃ < 6.458 and β₄ < 8.47) from Selberg weights that account for numbers with\nup to three prime factors (the three figures re-read in the thesis PDF on\n2026-09-11, each beside its κ and the three-prime-factor weights). That\nimproves on Franze's 4.516 and is still worse than DHR's 4.26645, so it adds a third independent point to the\nsuperlative in the status block above rather than disturbing it: at κ = 2 the\npublished field is Rosser–Iwaniec 4.834, Ankeny–Onishi 4.42, Λ²Λ⁻ 4.516,\nBlight 4.45, DHR 4.26645, and the exponent this note proves is the smallest of\nthem. The precise value is due to A. Booker & T. D. Browning,\n*Square-free values of reducible polynomials*, Discrete Analysis 2016:8,\narXiv:1511.00601, whose ancillary table (computed by interval arithmetic,\neach entry correctly truncated at the 20th decimal place) gives rigorously\n\n  β₂ = 4.26645028414864191641   (lower bound; add 10⁻²⁰ for an upper bound).\n\nShape of the theorem we invoke (the DH book's Theorem 9.1; conclusion quoted\nverbatim through Franze–Kao, arXiv:1812.11280, eqs. (19)/(20)): let A be a\nfinite integer sequence, and for squarefree d | P(z) suppose\n\n  |A_d| = (ω(d)/d)·X + r_d,     ω multiplicative,  0 ≤ ω(p) < p,\n\nwith the dimension condition in the one-sided product form\n\n  (Ω(κ,L)):  ∏_{z₁ ≤ p < z₂} (1 − ω(p)/p)^{−1} ≤ (log z₂ / log z₁)^κ · (1 + L/log z₁)\n             (2 ≤ z₁ < z₂),\n\nfor κ = 2. Then for any 2 ≤ z ≤ y, with u = log y / log z,\n\n  S(A, z) ≥ X · V(z) · { f₂(u) − O((log log y)² / (log y)^{1/(2κ+2)}) }\n            − 2 Σ_{m | P(z), m < y} 4^{ν(m)} |r_m|,\n\nwhere V(z) = ∏_{p<z}(1 − ω(p)/p), the error exponent is 1/(2κ+2) = 1/6 at\nκ = 2, and the DHR lower function f₂ vanishes on (0, β₂], increases\nmonotonically for u > β₂, and tends to 1, in particular f₂(u) > 0 for\nu > β₂. (The 2·4^{ν(m)} weighting at level y is the DH book's remainder form;\nthe older Halberstam–Richert condition R(κ,α) carries 3^{ν(d)}, and some\nformulations need only Σ|r_d|. We take the heaviest form since even it is\nharmless here, see §3.)\n\n**Our sieve problem.** Fix an interval (x, x+H] and set\nA = { r(r+2) : x < r ≤ x+H }, z = pₙ + 1, X = H. (Not z = pₙ: S(A, P, z)\nconventionally sifts the primes p < z, so z = pₙ would fail to sift pₙ\nitself; z = pₙ + 1 sifts all p ≤ pₙ, as required.) For squarefree d | P(z), the\ncondition d | r(r+2) confines r to exactly ω(d) residue classes mod d, where\n\n  ω(2) = 1,  ω(p) = 2 (odd p),  ω(d) = ∏_{p|d} ω(p) = 2^{ν'(d)}\n\n(ν'(d) = number of odd prime factors; the classes are distinct mod odd p\nbecause 0 ≢ −2). Counting each class in an interval of length H:\n\n  |A_d| = (ω(d)/d)·H + r_d,   |r_d| ≤ ω(d) ≤ 2^{ν(d)}.        (∗)\n\n**Dimension check.** By Mertens' theorem applied to ∏(1 − 2/p),\n\n  ∏_{z₁ ≤ p < z₂} (1 − ω(p)/p)^{−1} ≤ (log z₂ / log z₁)² · (1 + L/log z₁)\n\nwith an absolute constant L, the product-form condition Ω(κ,L) holds at\nκ = 2. (Equivalently, in sum form: Σ_{w≤p<z} ω(p) log p/p =\n2 Σ_{w≤p<z} log p/p + O(1) = 2 log(z/w) + O(1), the condition Ω₂(2) with an\nabsolute A₀; Halberstam–Richert *Sieve Methods* Lemma 5.3 is cited from secondary\naccounts as deriving the product bound from Ω₂(κ) + Ω₁ in general; not\nverified at the 1974 page.) Also 0 ≤ ω(p) < p holds: ω(2) = 1 < 2 and\nω(p) = 2 < p for odd p. None of this is exotic, and the honest description is\nthat the setup is quoted rather than built: the DH book's Example 1.2 (§1.3\n\"Prime g-tuples\", pp. 7–8) is L(n) = ∏_{i≤g}(a_i n + b_i) taken **on an\ninterval**, A = {L(n) : x − y < n ≤ x} with X = y, ω(d) the number of\nincongruent solutions of L(n) ≡ 0 mod d, |r_A(d)| ≤ ω(d) and ω(d) ≤ g^{ν(d)}.\nEverything in this paragraph and the preceding one is that example at g = 2,\nL(n) = n(n+2), Δ = 2. The density product is\n\n  V(z) = (1/2) ∏_{2<p<z} (1 − 2/p) ~ (2C₂ e^{−2γ}) / log² z,\n\nwith 2C₂e^{−2γ} = 0.41621…, the constant verified numerically in this\nrepository (research/genealogy.js: δ·ln²p → 0.4150 at p = 9973 against\n0.41621). So V(z) ≍ 1/log²z: genuinely dimension 2, and the linear sieve\n(with its sifting limit 2) is unavailable. This is the precise\ntechnical content of \"the twin problem is two-dimensional\" (cf. FKMPT,\nJ. Eur. Math. Soc. 23 (2021), 667–700, Remark 7, verified in the arXiv\nversion 1802.07604, the journal page of the remark not checked; corrigendum\nibid. 25 (2023), 2483–2485).\n\n## 3. The interval application, and why the remainder does not explode\n\nThe directive-level worry: with two classes per prime, the per-divisor\nremainder is 2^{ν(d)}, not ≤ 1 as in Iwaniec's one-class setting, and the\nDH remainder form of Theorem 9.1 weights it by another 4^{ν(m)}, times 2.\nDoes the remainder sum swamp the main term? It does not, and as far as we\ncan see this is the only reason the note is easy where Iwaniec's theorem was\nhard:\n\n  2 Σ_{m < y, m | P(z)} μ²(m) 4^{ν(m)} |r_m|\n    ≤ 2 Σ_{m < y} μ²(m) 4^{ν(m)} 2^{ν(m)}\n    = 2 Σ_{m < y} μ²(m) 8^{ν(m)}\n    ≪ y (log y)⁷,\n\nby the standard mean value of k^{ν(m)} (Σ_{m≤Y} μ²(m) k^{ν(m)} ≍\nY (log Y)^{k−1}, here k = 8); an elementary form suffices, since\nμ²(m) 8^{ν(m)} ≤ τ₈(m) and Σ_{m≤Y} τ₈(m) ≤ Y (1 + log Y)⁷ by counting ordered\nfactorizations. Polynomial in y with a polylog, an ε in the\nexponent absorbs it entirely.\nIwaniec had no ε to spend: at u = 2 exactly, every log matters, which is why\nhis proof needs the refined error analysis of the linear sieve. At u = β₂ + ε\nwe are strictly inside the positivity region and can be wasteful.\n\n**Assembling.** Choose H = z^{β₂+ε} and level y = z^{β₂+ε/2}. Then u =\nlog y/log z = β₂ + ε/2 > β₂, so f₂(u) ≥ f₂(β₂+ε/2) =: c(ε) > 0 (f₂\nincreasing), and the explicit error O((log log y)²/(log y)^{1/6}) is < c(ε)/2\nfor z ≥ z₀(ε), so:\n\n- main term:  H · V(z) · { f₂(u) − O(·) } ≫ c(ε) · z^{β₂+ε} / log² z,\n- remainder:  ≪ y log⁷ y ≪ z^{β₂+ε/2} · log⁷ z.\n\nThe main term dominates by the factor z^{ε/2}/log⁹z → ∞. Hence S(A, z) > 0:\n**every interval of length z^{β₂+ε} contains a twin candidate**, uniformly in\nthe interval's position x (uniformity is free: (∗) holds for every x with the\nsame constants, and the O(·) in Theorem 9.1 depends only on the Ω-condition\nconstants). Taking x to range over a period gives G₂(n) ≪_ε pₙ^{β₂+ε}; the\nfinitely many n with pₙ < z₀(ε) are absorbed into C(ε). ∎ (modulo §6)\n\n## 4. No transfer lemma needed\n\nIwaniec's 1978 paper needs its Lemma 1, the divisor-bijection transfer\ncarrying the primorial estimate to arbitrary squarefree moduli, and that\nlemma is precisely the step queried in the unanswered\n2016 MathOverflow question 245539. One unanswered post is not a controversy and\nthe lemma is not known to be wrong; what would help is an explicit-constant or\nformalised exposition. **The argument avoids it entirely**: G₂ is defined at\nprimorials, the sifting set is \"all primes ≤ pₙ\", and the sieve above is run\ndirectly there. (For general squarefree q the analogous statement with z =\nP⁺(q) + 1 follows by the same argument sifting only p | q, the dimension\ncondition Ω₂(2) holds a fortiori with the same constants, but the resulting\nbound is in terms of P⁺(q), not ω(q); the sharper ω(q)-form for general q is\nexactly where a Lemma-1-style transfer would be needed, and we make no claim\nthere.)\n\n## 5. Numerical sanity, and the bracket the truth sits in\n\nThe bound versus the verified data, at the largest computed level (pₙ = 43):\n\n  bound (ignoring C(ε)): 43^{4.26645} ≈ 9.3 × 10⁶;  actual G₂ = 618.\n\nSlack of four orders of magnitude, a factor of 1.5 × 10⁴, and the data\ncannot say how much of it is real. *(Until 2026-09-09 this read \"(pₙ = 41):\n41^{4.26645} ≈ 7.6 × 10⁶; actual G₂ = 546\", a factor of 1.4 × 10⁴; the\nfourteenth term was in the repository's ladder since 2026-08-18 and had not\nbeen carried into this note.)* *(Until 2026-08-18 this read \"at the largest\ncomputed level (pₙ = 37): 37^{4.26645} ≈ 4.9 × 10⁶; actual G₂ = 528\", a factor\nof 9.3 × 10³. The new level widens the gap, as it must while the truth sits\nnear exponent 1.5 and the bound at 4.27.)* On ten terms of G₂ (pₙ in [5, 37]) a power fit in pₙ returns 1.801 ± 0.074\n(research/exponent-control.js, OUTPUT table S7), and that number is not the\nexponent: the same estimator run on 58 terms of the one-class Jacobsthal\nfunction returns 1.282 ± 0.008 with white residuals and no drift\n(research/exponent-control.md §1), against a one-class exponent that is\nconjecturally 1 (Maier and Pomerance, g(x#) = x (log x)^{2+o(1)}) and proven\nonly to lie in [1, 2] (the FGKMT lower bound quoted below, Iwaniec's upper\nbound g(x#) ≪ x²; Erdős problem #687 asks for o(x²)). The correction that\nfollows assumes the conjectured value for the control. On that assumption the\ncontrol's bias gives 1.54 ± 0.09 for G₂ (the equal-bias column of the same\ntable, 1.539 ± 0.094) and 1.57 ± 0.06 for the dominating h₂ of Ziller and\nMorack (1.566 ± 0.058 there), whose 19 terms give the longer lever. **Central estimate\n1.57, practical bracket 1.3 to 1.9**, with a proven floor of 1 (h₂ ≥ h, and\nh(x#) ≫ x log x logloglog x / loglog x by FGKMT, quoted below) and exponent 2\ndisfavoured by the one-sided direction of\nthe control's bias rather than excluded by the data. *(Update, 2026-08-21: the\nfit has since been re-run on all 22 trusted terms of A144311, pₙ ≤ 79, against\nthe 64-term control: raw 1.777 ± 0.029, corrected central **1.50 ± 0.05**\nstatistical with the systematic unquantified, practical bracket 1.3 to 1.8;\nh₂'s 1.57 ± 0.06 is unchanged, its ladder did not extend. The ten-term\nfigures above stand as this note's original record;\n`research/exponent-control.md` §5.)*\n\nThe object is now bounded from below as well, and that side is free. Twin slots\nare a subset of the holes of the\nsame tile, so G₂(x#) ≥ g(x#) pointwise for the ordinary Jacobsthal function g,\nand the Rankin–Pintz–Ford-Green-Konyagin-Maynard-Tao machinery transfers\nunchanged:\n\n  G₂(x#) ≫ x · log x · logloglog x / loglog x   (PROVEN, by monotonicity;\n  research/two-class-lower-bounds.md §3).\n\nPer the audit this is the first lower bound of any kind recorded for a\ntwo-class Jacobsthal function.\n\nThe point of this note is not sharpness. The interval of provable exponents was\nentirely empty before, in both directions; the theorem above fills it at 4.267,\nthe Ziller–Morack-style conjectural ceiling sits at 2 (h₂(n) < pₙ² − pₙ; a\nbound G₂(x#) < x′² − 2 with x′ the prime after x, exponent 2 with constant\nbelow 1, would by the p²-rule imply the twin prime conjecture,\nresearch/G2-STATE.md §1c and §5, and a constant at exponent 2 does not turn\ninto that on its own), and the open band is therefore (2, β₂] = (2, 4.26645…].\n\n## 6. Every step not fully justified here\n\n1. **The DHR theorem, verified against the primary source (2026-08-14),\n   no outstanding items.** Read directly from the Diamond–Halberstam book:\n   Theorem 9.1 (pp. 103–112); the formal **Ω(κ) condition, Definition 1.3,\n   eq. (1.5), p. 8**, ∏_{w₁≤p<w}(1−ω(p)/p)⁻¹ ≤ (log w/log w₁)^κ(1+A/log w₁)\n   **for all pairs 2 ≤ w₁ < w**, the exact product form this note invokes\n   (their A = our L); Theorem 6.1\n   (pp. 67–68); β₂ ≈ 4.266 (p. 79). Every element confirmed: 2·4^{ν(m)}\n   remainder, 1/6 error exponent, S sifting p<z, the difference-differential\n   f_κ, the positivity mechanism (f_κ>0 ⟺ u>β_κ; below it the sieve is trivial,\n   p. 79 verbatim), β₂ ≈ 4.266 as the smallest published κ=2 sifting limit\n   (§2), and\n   α_κ ≥ β_κ+1 for κ≥2 (p. 77).\n   **The book does our setup and our density check for us.** Its own §1.3–1.4\n   motivating example (Example 1.2, pp. 7–8) is A = {L(n) : x − y < n ≤ x},\n   already on an interval, with X = y: it defines ω(d) = #{incongruent\n   solutions of L(n)≡0 mod d}, notes ω(p) ≤ g with equality for p∤Δ, bounds\n   |r_A(d)| ≤ ω(d) ≤ g^{ν(d)}, and states (p. 8) \"Ω(κ) holds in Example 1.2 with\n   κ = g.\" For L(n)=n(n+2): g=2, Δ=2, so ω(p)=2 for odd p (the two roots\n   n≡0, n≡−2) and ω(2)=1, *precisely our tile's forbidden classes*, giving\n   dimension κ=2. The book's check uses only ω(p) ≤ g and nothing about the\n   polynomial, so it holds at every dimension. The density hypothesis §2\n   verifies by hand is the book's own worked example, and so is the sequence.\n   No mathematical item remains open on the sieve input; the theorem rests on\n   primary sources for the book's part, on secondary accounts for the\n   Halberstam–Richert 1974 page (§2), and its constants are inexplicit (§6.4).\n   What is ours is the dimension-2 instantiation, not the method: reading a\n   Jacobsthal bound off a sieve's error exponent is on record at dimension one\n   (MathOverflow 37679, answer 52890, 2011, j(x#) ≪ x^{4.032};\n   research/G2-STATE.md §8), and the closeness of the two exponents is\n   coincidence.\n   Audit trail: research/dhr-verification.md.\n   **Definition 1.3 is sourced primarily and corroborated twice.** The\n   quantifier over all pairs (w₁, w) is the load-bearing part: a κ that holds\n   only at some pairs, or only by letting A grow, is not a dimension. The book's\n   page carries it, together with the constants κ ≥ 1, A > 1. Two independent\n   published restatements agree verbatim, and are kept because they were reached\n   first and because a second reader's transcription is worth having:\n   **D. R. Johnston and S. N. Thomas, *The sum of a prime power and an\n   almost prime*, arXiv:2503.04045v3 (14 May 2025), §2.3**, which prints\n   ∏_{z₁≤p<z₂, p∈P}(1−g(p)/p)^{−1} < (log z₂/log z₁)^κ{1 + L/log z₁}, **\"for\n   z₂ > z₁ ≥ 2\"**, citing \"Diamond, Halberstam, and Galway, *A Higher-\n   Dimensional Sieve Method*, CUP 2008\", Ch. 11.1; and **K. Ford, *Sieve methods\n   lecture notes*, Spring 2023, p. 18**, which gives the same inequality for\n   2 ≤ y ≤ w ≤ z and adds that κ is \"the smallest admissible value … with B\n   remaining bounded\". Note Johnston–Thomas restrict the product to p ∈ P, the\n   sifting set, which the book's own eq. (1.3) licenses (ω(p) = 0 off P). Our\n   ω(p) = 2 for odd p ≤ pₙ satisfies the condition at every pair with an\n   absolute A, which is §2's dimension check and is unaffected.\n2. **The remainder form, verified.** Theorem 9.1 accepts remainders through\n   2 Σ_{m|P(z), m<y} 4^{ν(m)}|r_m| at level y (not the older\n   Halberstam–Richert 3^{ν} condition, and not a bilinear/well-factorable\n   structure). §3 has been re-run against this actual form (8^{ν}, y log⁷y):\n   the conclusion is unchanged, the bound beating the requirement by z^{ε/2},\n   exactly the outcome an earlier draft of this section predicted for any\n   standard remainder convention.\n3. **The o(1) in the sieve's main term, verified and explicit:** it is\n   O((log log y)²/(log y)^{1/(2κ+2)}) = O((log log y)²/(log y)^{1/6}) at\n   κ = 2, uniform given the Ω-condition constants (Franze–Kao's restatement\n   of Thm 9.1). The precise dependence of the implied constant on the\n   Ω-condition constants is not made explicit anywhere we have read, which is\n   the same inexplicitness item 4 records for C(ε); nothing in §3 needs it,\n   since the main term beats the remainder by z^{ε/2}.\n4. **Constants are inexplicit** (as in Iwaniec's own theorem, where the constant\n   at Erdős #970 is not known). A fully explicit version would need\n   explicit dimension-2 sieve bounds (possibly via Franze's Λ²Λ⁻ with the\n   worse exponent 4.516 but explicit machinery).\n5. **Fallback if the DHR citation fails:** the Fundamental Lemma of sieve\n   theory. In the Friedlander–Iwaniec *Opera de Cribro* Lemma 6.8\n   normalization (quoted verbatim as Lemma 9.1 of Matomäki–Teräväinen,\n   arXiv:2301.07679), the level D = z^s requires **s ≥ 9κ + 1 = 19** at\n   κ = 2, with main-term positivity factor 1 − e^{9κ−s}K^{10}, so positivity\n   needs s > 9κ + 10 ln K, where K is the constant of the lemma's hypothesis\n   (iii): ∏_{w₁≤p<z₁}(1 − h(p))^{−1} ≤ K (ln z₁/ln w₁)² for all\n   z₁ ≥ w₁ ≥ 2. For the twin sequence sifted by every prime, h(3) = 2/3\n   forces K ≥ 3 (take w₁ = 3 and let z₁ decrease to 3), so the exponent as\n   written is **18 + 10 ln K + ε ≥ 28.98 + ε**, not 19 + ε (return #26,\n   2026-09-11). The exponent 19 + ε is recovered by fixing the residue class:\n   sieve A′ = {r(r + 2) : r ≡ a (mod W), x < r ≤ x + H}, with\n   W = ∏_{p<23} p = 9 699 690 and a a class with (a(a + 2), W) = 1, by the\n   primes 23 ≤ p ≤ pₙ only. Then h(p) = 0 for p < 23, |r_d| ≤ 2^{ν(d)} still\n   holds with X = H/W, and the constant is\n   K(23) = sup_{z ≥ w ≥ 23} ∏_{w≤p<z}(1 − 2/p)^{−1} (ln w/ln z)² =\n   1.1039848905…, the limit at the block {29, 31}, certified for every z by\n   an exact scan of all prime pairs below 10⁶ and Rosser–Schoenfeld's\n   Theorem 5 beyond (return #166, the return for job #371, `k-certificate.py`;\n   re-run for this revision, output identical). Since K(23) < e^{0.1},\n   s₀ = max(19, 18 + 10 ln K(23)) = 19 and G₂(pₙ#) ≪_ε pₙ^{19+ε}, the\n   absolute factor W absorbed in the constant. The smallest admissible\n   modulus is this one: w₀ = 19 gives K(19) ≥ 19/17 > e^{0.1}. Formulations\n   differ in the constant (HR-1974-type forms give positivity at an absolute\n   but inexplicit u₀(κ)); any such version yields the same theorem at some\n   finite exponent.\n6. **Ceiling acknowledged:** exponent 2 with constant below 1 (G₂(x#) <\n   x′² − 2) implies the twin prime conjecture by the p²-rule, and the\n   zone-occupancy weak form is equivalent to it (research/G2-STATE.md §1c); a\n   bound at exponent 2 with an unspecified constant implies neither. That\n   threshold is not reachable by known sieve methods alone (parity; Selberg's\n   examples). Improving 4.266… toward 2 is\n   the recognized dimension-2 sifting-limit problem. Nothing in this note\n   moves the wall; it fills the empty shelf in front of it.\n\n## 7. What would make this publishable\n\nThe sieve input is verified against the primary source and the derivation is\nclosed (§6). What remains is presentation and risk control. (i) Add the\nexplicit-constant variant via Franze's tables; (ii) state the\ngeneral-squarefree-q corollary in terms of P⁺(q); (iii) a referee-proof rewrite\nof (∗) (the mean-value estimate now has its elementary form in §3; the lower\nbound is already folded in, §5);\n(iv) an expert sanity pass, since the\nresult is modest enough that the main risk is not depth but a convention\nmismatch in the sieve statement. Companion citations: A059861 (census), A144311 (the ladder, as G₂ − 1;\nthe OEIS draft research/oeis-G2-submission.md is closed as a duplicate of it),\nErdős, *On the integers\nrelatively prime to n and on a number-theoretic function considered by\nJacobsthal*, Math. Scand. 10 (1962), 163–170, for the one-class ancestor,\nZiller–Morack for the h₂ contrast, FKMPT Remark 7 for the two-dimensionality\ncontext, Granville's *Sieving intervals and Siegel zeros* for the one-class\nmechanism this note deliberately does not need.\n\n## 8. Authorship & AI disclosure\n\nSole author: Chris Benjaminsen.\n\n> The framework, vocabulary, and driving questions are the author's,\n> developed over six years of independent work. Formal derivations,\n> literature audits, computations, and manuscript drafting were carried out\n> using AI assistants under the author's\n> direction. Computations have reproducible code and recorded outputs;\n> asymptotic arguments require their stated mathematical inputs and are not\n> proved by finite checks. Refuted intermediate claims are retained in the\n> record.\n"}