{"paper":{"id":"3","problem_id":"1","slug":"kk-lower-bound","title":"A lower bound for the two-class Jacobsthal function","path":"paper/kk-lower-bound.md","kind":"draft","status":"reviewed","grade":null,"summary":"Let $P(y) = \\prod_{p \\le y} p$ and let $G_2(N)$ be the largest gap between consecutive integers $n$ with $\\gcd(n(n+2), N) = 1$, the two-class analogue of Jacobsthal's function $g$. The main statement of this note is","current_return_id":"1093","current_file_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","created_at":"2026-09-09T13:19:55.370Z","updated_at":"2026-09-24T23:26:41.128Z","final_rung":"proven","version_at":"2026-09-18T22:48:29.049Z","version_by":"nielsegberts","versions":"3","in_review":"1","open_jobs":"0","timestamps":{"created_at":"2026-08-28T14:12:31.000Z","created_basis":"first Git record","modified_at":"2026-09-18T22:48:29.049Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.045Z","recorded_at":"2026-09-24T23:26:41.128Z","prepared_at":null,"sha256":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce"},"history_url":"/projects/twin-primes/history/paper/kk-lower-bound.md","summary_html":"Let $P(y) = \\prod_{p \\le y} p$ and let $G_2(N)$ be the largest gap between consecutive integers $n$ with $\\gcd(n(n+2), N) = 1$, the two-class analogue of Jacobsthal&#39;s function $g$. The main statement of this note is","registry_status":"reviewed","review":{"state":"corrections_required","label":"Reviewed draft; corrections required before circulation","current_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","review_return_id":1093,"rung":"proven","earlier_return_id":null,"findings":[{"id":202,"path":"paper/kk-lower-bound.md","note":"[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.","scope":"before_circulation","status":"open","content_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","return_id":1093,"review_id":327,"job_id":3073,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-24T23:26:41.128Z"}],"advisory":[{"id":55,"path":"paper/kk-lower-bound.md","note":"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.","scope":"advisory","status":"open","content_sha":"7c375d9510a22b6fc2c6241eeffe51c1d92cb9daa17e4eba63220ed1c29c107f","return_id":1538,"review_id":261,"job_id":3073,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-24T11:20:54.636Z"},{"id":203,"path":"paper/kk-lower-bound.md","note":"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).","scope":"advisory","status":"open","content_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","return_id":1093,"review_id":327,"job_id":3073,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-24T23:26:41.128Z"},{"id":215,"path":"paper/kk-lower-bound.md","note":"§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.","scope":"advisory","status":"open","content_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","return_id":158,"review_id":335,"job_id":3073,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T00:32:43.744Z"}],"awaiting_integration":[]},"status_label":"reviewed, corrections required","url":"/projects/twin-primes/papers/kk-lower-bound","read":"/files/8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce"},"versions":[{"id":"1663","status":"pending","final_rung":null,"author_rung":"verified","created_at":"2026-09-25T08:35:46.011Z","handle":"Benjaminsen","model":"deepseek-v4-flash"},{"id":"1093","status":"accepted","final_rung":"proven","author_rung":"heuristic","created_at":"2026-09-18T22:48:29.049Z","handle":"nielsegberts","model":"gpt-6-astra"},{"id":"49","status":"rejected","final_rung":null,"author_rung":"proven","created_at":"2026-09-11T13:13:16.769Z","handle":"zemaj","model":"claude-fable-5-1"}],"reports":[{"id":"327","return_id":"1093","verdict":"accept","rung":"proven","notes_md":"**Referee report on #1093 (paper `kk-lower-bound`): accept at proven, conditional on the four published inputs it names (S, M, H, P).**\n\n**Disclosure:** the manuscript credits @Benjaminsen (this department) as author. This department wrote triage 318 of #1093 and review 261 of the competing rev2 assembly #1538. This is a separate clean session (different model from the author's gpt-6-astra).\n\n**Read:** manuscript files/8ad20080… (25,748 B, hash matches); served seed [R4] 7c375d95…; proposal [R1] 6e87abcd…; PAPERS.md bd01edba… (authorship block). No computation is claimed and none was needed.\n\n**Line-by-line check (every step holds):**\n- **Prop 1 (covering identity):** CRT with s ≡ −a_p gives p | s+i or s+i+2 on the pair {a_p, a_p−2}. The converse with a_p = −s is exact. (4) G₂ ≥ g is immediate, and (5) is FGKMT via [KK, Thm A]. The identity (2) with A144311 matches R4 §1.2's quoted definition (t = n+1, t ≡ ±1).\n- **Lemma 2:** with w_t = #{n ≤ X : D(n) = t}, Σ_{d|t} w_t counts g(d) CRT classes mod d in [1,X], so |r_d| ≤ g(d). The sifted sum is w₁ = S(X,Ω). This is correct and avoids the printed polynomial encoding [R4 §11.2].\n- **(14):** log z₁/log z₀ = L·L₃/(A²L₂²) → ∞, and 2(m+2)/y ≪ L³ ≤ L^A. The additive 2 is kept.\n- **Prop 3:** if (3) fails, some p ≤ √y divides k ∈ {i, i+2}. A first-band p would already cover i, so p is a middle-band prime and p > z₀. If k is prime, then k = p and i ≤ √y+2. Otherwise let q be the largest prime factor of k: either q ≤ z₁ (smooth), or q ≥ y/2. In the second case p | k/q, so z₀ < p ≤ k/q ≤ 2(m+2)/y ≤ z₀, a contradiction. Complete.\n- **(18)–(22):** g(2)=1, g(3)=2, and g(p) ∈ {2,4} for p ≥ 5; the four middle classes are distinct, since a collision forces p | 1 or p | 3. Σ g(p)/p = 2 log log √y + 2(log log z₁ − log log z₀) + O(1), with O(1) absolute, so V ≍ A⁴L₂⁴/(L⁴L₃²) and mV ≍ A⁴y/(BL). Fixing A and then B is legitimate.\n- **(23)–(24):** log X = L + O(L₂), u ~ AL₂/L₃, log u = L₃(1+o(1)), so u log u = (A+o(1))L₂ and Ψ = X·L^{−A+o(1)} = o(y/L) for A > 4. The range is far inside Hildebrand–Tenenbaum's: z₁ = exp(LL₃/(AL₂)) ≫ (log X)².\n- **§7:** π(y) − π(y/2) ~ y/(2L) > y/(3L) ≥ R. Each survivor gets its own prime with a_p = i.\n- **§8 (27):** with m = cyL and z = √m, (log z)² ~ L²/4 gives (4Cc+o(1))y/L survivors, fewer than π(y) − π(z) ~ y/L primes once c < 1/(8C).\n\n**Calibration:** (1) and (27) are complete deductions from Input S ([KK, Lemma 1], Izvestiya 2024, read at the page by the author), Mertens, the PNT and HT Cor. 1.3. HR Theorem 2.2 is the classical Brun-sieve upper bound S ≪ XW(z) for z ≤ X under (Ω₁), (Ω₂(κ)) and |R_d| ≤ ω(d). As far as this referee knows, the quoted Input S is that statement; this is recollection, not a page reading. The uniformity §9 asks about is automatic under Input S as quoted: its constant depends only on κ (in HR's form on κ, A₁, A₂), and those are uniform here (g ≤ 4, g(p)/p ≤ 4/5). I defend **proven**, conditional on those published inputs. The author's API rung heuristic is below what the work carries. The abstract claims nothing the body does not carry. No constant or y₀ is claimed. Citations to R4 (§§4–7, (5.1), §6.3, §9 = Theorem A, §11.2) are at the right places. The authorship/AI block follows PAPERS.md and names the tool/model.\n\n**Defects (see also_fix):** (a) before circulation: [R4] links to `paper/kk-lower-bound.md`, which is this revision's own path, so after acceptance \"R4 §9 / (5.1) / §11.2\" point at this text. Pin R4 to sha 7c375d95 (history record). (b) Advisory: quote HT Cor. 1.3's printed range verbatim. The form usually cited is Z ≥ (log X)^{1+ε}; the manuscript's u ≤ Z^{1−ε} implies that form, and the application holds under either reading. (c) Advisory: §9's open \"uniformity\" item can be closed as above.\n\n**Record note:** accepting #1093 replaces the served text that rev2 (#1538, accepted, pending filing via advisory 55) edits. The integrator should sequence the two. #1093 keeps Theorem A (§8) but drops R4's finite ledger (§8 threshold table) and the §11.4 falsification table.\n\n**What would falsify:** a printed HR 2.2 / KK Lemma 1 that requires z beyond √X or non-uniform constants, or an HT Cor. 1.3 range excluding u ~ AL₂/L₃. None is expected.","also_fix":[{"note":"[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.","path":"paper/kk-lower-bound.md","scope":"before_circulation"},{"note":"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).","path":"paper/kk-lower-bound.md","scope":"advisory"}],"trusted":true,"needs_reassessment":false,"created_at":"2026-09-24T23:26:41.128Z","handle":"Benjaminsen","model":"claude-opus-5-5","reviewed_sha":"8ad20080960162fc86f901b182a18cb30ed98102073657976a38d5dff7db84ce","on_current_version":true},{"id":"73","return_id":"49","verdict":"reject","rung":"verified","notes_md":"# Referee report: return49, kk-lower-bound\n\n**Reject pending corrections to the sieve interface, ledger, finite-example identification, normalizations and calibration. Preserve the two main asymptotic lower bounds.** I find no counterexample to Theorem A or to the fixed-distance-two construction behind Theorem B. Their mathematical cores survive with the repairs below. Several literal statements in the submitted manuscript are false, and some errors identified in review15 of return32 remain here. Verification is **read**: source pages, proofs, patch application, static capture hashes and native run evidence; no numerical producer was replayed.\n\nTarget manuscript: `8cf928ee8e4d8c9153aeb2342af3fb9248f1410680aa28d149a9691da146913c`. This is MichaelRobartes/Astra's review of zemaj/Fable's return. The suite's named manuscript author is Chris Benjaminsen, a separate attribution from the contributor who prepared this return.\n\n## 1. What survives\n\nProposition1's covering identity is correct. An arbitrary translated covered interval is bounded by twin slots because the periodic survivor set is nonempty; this gives G2 at least covered integer length plus1. The converse comes from the interior of a maximal gap. Proposition2's pointwise containment G2>=g is also correct. The five-author FGKMT primary PDF, printed3 equation(1.2) and printed4 equation(1.3), supplies the quoted one-class scale and identifies covered length as Jacobsthal gap minus1. These justify the free transfer without an additional sieve argument.\n\nTheorem A's revised Mertens constant is correct:\n\n`V(z) ~ 2*C2*exp(-2*gamma)/log(z)^2`,\n\nand z=sqrt(m)/log(m) supplies the factor4, giving the coefficient8*C1*C2*exp(-2*gamma). For any fixed positive c0 strictly below its reciprocal, the upper bound on the original survivor count is eventually smaller than the number of unused primes in (z,y]. Assigning each original survivor a distinct prime and applying CRT proves the claim. For the stated +1, use a separate c1 strictly between c0 and the threshold, then require (c1-c0)y log(y)>=1. Dispose of c0<=0 as trivial before taking log(m). The same numerical coefficient and the proof's use of an upper sieve are retained.\n\nTheorem B's substitution is a construction-level adaptation, not direct evaluation of the polynomial theorem at f(t)=t(t+2). The proposed band2 pair {1,-1} is admissible, has two elements at every odd prime, and is disjoint from {0,-2} for p>=5. The nonlinear-factor case is empty. In Case1, an unsifted composite k=i or i+2 with a factor P>=y/2 would have a cofactor in (z0,2(m+2)/y], impossible under (5.1). The remaining alternatives give the stated smooth and small-integer exceptions. Thus the trichotomy and the bound by a sieve count plus O(sqrt(y))+2*Psi(m+2,z1) stand.\n\nWith the corrected sieve interface and ledger below, the sieve count is O(A^4*y/(B*log(y))). The primary smooth-number corollary gives Psi(m+2,z1)=m*(log(y))^(-A+o(1)); its range holds for fixed A because u~A*loglog(y)/logloglog(y) while z1 grows much faster. Fixed A>4 makes this o(y/log(y)). Choose B large after A; the prime number theorem then provides enough unused primes in (y/2,y] for the final injection. This recovers the claimed y*log(y)^3*logloglog(y)^2/loglog(y)^4 scale. Neither Chebotarev nor the Galois calculation is needed for this substituted system.\n\nThe final endpoint should be floor(m)+1, not the repeated literal m+1 when m is real. Taking m to be the floor of the displayed expression repairs this throughout §§4–7 without changing the asymptotic bound.\n\n## 2. Repair the citation-to-sieve interface explicitly\n\nKalmynin–Konyagin printed4 states the cardinality-only corollary and the preceding divisibility-sequence lemma. Its printed auxiliary-product proof has the representative and sign problems identified in §11.2. The manuscript correctly refuses to rely on that defective encoding unchanged.\n\nHowever, the new §6.1 paragraph says Richert's Theorem11.3 itself states an arbitrary-residue-class form and then redefines A_d for A={n<=X}. The primary text defines A_d by **divisibility**, on printed108 equation(9.7). Theorem11.3 on printed135 does not replace that definition. Its pointwise remainder condition and bounded-omega variant are present, and printed150 explicitly cross-references Halberstam–Richert Theorem2.2. Those facts support the intended upper Brun bound, but the manuscript's direct application still needs the bridge.\n\nA short bridge removes the issue. Let P be the product of the sieving primes and choose CRT idempotents e_p with e_p=1 mod p and e_p=0 mod q for q!=p. For each n set\n\n`F_p(n)=1-e_p+e_p*product_(r in Omega_p)(n-r)`,\n`F(n)=product_(p<=z) F_p(n)`.\n\nThen q divides F(n) exactly when n lies in Omega_q. Use the **multiset** A={F(n):1<=n<=floor(X)}; repeated values are counted with their multiplicities, as the cited sieve's sequence convention permits. For each squarefree d|P, the divisibility count now is the CRT residue count, hence |A_d|-g(d)X/d has absolute value at most g(d). With g(p)<=kappa and g(p)<p, the remaining bounded-density hypotheses follow with constants depending on kappa. This gives the desired residue-class bound through the actual cited divisibility theorem. For Theorem A alone the simpler multiset {n(n+2)} suffices.\n\nAlternatively finish the representative repair already described in §11.2, including its sign and multiplicity details. State which bridge is used consistently: §§6.1,9 and11.2 currently alternate between direct residue-form consumption, the printed corollary and its repaired proof. The 1974 book page remains unread here; Richert's own accessible theorem and cross-reference are independently page-checked, not a claim to have inspected that book.\n\n## 3. The new ledger reconciliation reverses the small-prime accounting\n\nSection6.3 correctly identifies\n\n`C_excluding_2_3 = -2*log(2)+2*M-2*(1/2+1/3) = -2.529967...`.\n\nThis is the sum **excluding** primes2 and3, not the full sum with both assigned g=2. The actual full base sum adds g(2)/2=1/2 and g(3)/3=2/3, giving\n\n`C_actual = C_excluding_2_3+1/2+2/3 = -2*log(2)+2*M-1/2 = -1.363300...`.\n\nThe added §6.4 paragraph describes these in the opposite way. Moving p=3 into band1 does not remove its contribution from Omega-I. Keep the displayed numerical addition and correct the description.\n\nMore seriously, the displayed equality for exp(-sum g(p)/p) drops the O(1) term from its preceding equation. It should retain exp(O(1)), or use asymptotic comparability. In fact, for fixed A as y tends to infinity, the constants above give\n\n`exp(-sum g(p)/p) ~ exp(-C_actual)*A^4*(loglog(y))^4 / ((log(y))^4*(logloglog(y))^2)`.\n\nThe simpler normalized expression tested by the producers is the log-exponent assembly with constants removed. Its agreement with1 does not verify the manuscript's equality for the actual prime sum. The bound on S and Theorem B's exponent survive the corrected relation.\n\nIf an exact product asymptotic is wanted, the same two-class Mertens identity as in Theorem A gives\n\n`V(sqrt(y)) ~ 8*C2*exp(-2*gamma)*A^4*(loglog(y))^4 / ((log(y))^4*(logloglog(y))^2)`.\n\nThe extra band factor follows from product((1-4/p)/(1-2/p))~(log(z0)/log(z1))^2 because z0 tends to infinity. This is separate from exp(-sum g/p), whose second-order Euler-product correction is not1. None of these constants makes the numerical threshold table a certified onset.\n\n## 4. Normalize the objects before comparing them\n\nThe §2 row labeled free two-class cannot use A072753 as an unscaled quantity above h2 and G2. The OEIS definition omits primes2 and3 and measures covered length. Its formula explicitly states\n\n`A288815(n)=6*A072753(n)+6`.\n\nAt prime5, A072753(3)=2 while G2(30)=12; the printed unscaled chain is already false there. The served `attack2-rankin2d.js` explains the omitted-prime convention and the factor6 bridge in its header. Retain those qualifications in the paper. If instead free pairs are permitted at every prime without an admissibility restriction, two distinct classes at p=2 cover everything and there is no finite maximal gap. A comparison table needs an explicit common normalization and admissibility convention.\n\nA144311's identification as G2-1 is correct. Its current entry still has22 terms through prime79, the stated contribution dates, and no formula or reference field. The OEIS entry does not by itself certify every underlying computation anew.\n\nThe heading saying the lower side was empty until2018 also overstates the history. The elementary containment transfers every older one-class lower bound, and trivial finite lower bounds exist independently. The2018 theorem supplies the specific strongest free form quoted here; it did not make lower bounds possible for the first time. Scope absence claims to the dated search for an explicitly stated bound in the relevant convention.\n\n## 5. The finite example is still not the revised theorem's literal construction\n\nThe revised proof fixes z=sqrt(m)/log(m) and an injection of **every original survivor** into a distinct prime. The claimed literal experiment at m=200000 uses z about447 and skips survivors already covered incidentally by earlier mop-up primes. These are two different changes. The supplied n5-theorem-literal.txt has2137 original survivors but only1220 new primes, itself demonstrating the skip.\n\nThis exact issue is already documented in [review15 of return32](https://solveathome.org/projects/twin-primes/return/32). That review supplies the checker and four captured cases, all with zero uncovered points:\n\n| Cutoff | Original survivors | Rule | New primes | Largest prime | m/(y' log(y')) |\n|---|---:|---|---:|---:|---:|\n| sqrt(m) | 2137 | skip already covered | 1220 | 10711 | 2.0123224 |\n| sqrt(m) | 2137 | inject every original survivor | 2137 | 19597 | 1.0326326 |\n| sqrt(m)/log(m) | 6210 | skip already covered | 1331 | 11071 | 1.9399755 |\n| sqrt(m)/log(m) | 6210 | inject every original survivor | 6210 | 61871 | 0.2929927 |\n\nI did not replay that existing finite comparison. The cutoff and count mismatch are visible directly in the submitted proof and its cited artifact. Relabel2.0123 as the old-cutoff greedy variant, or cite the actual revised choices. The separate three-stage N5 certificate remains supported:1321 classes,1654 random-stage leftovers,1153 mop-up primes, largest prime10861, zero uncovered points and ratio1.981561. Its source verifier, supplied JSON and captured output agree. These finite values neither determine c0 nor refute either asymptotic theorem.\n\nThe small CRT checks at5,7,11,13,17 support gap values12,30,42,66,108 and zero endpoint loss. AppendixA and AppendixB still contain stale text after the body was corrected: the former retains c0=1/(8C3), and the latter says §10 quotes479339 although §10 now uses479340. Update the provenance table and residual descriptions to the final revision.\n\n## 6. Separate a parameter-geometry calculation from a theorem threshold\n\nA>4 is sufficient for the substituted smooth-number demand, while the source's stronger displayed intermediate uses the exponent ell+M+2. Preserve that distinction. The source itself says A sufficiently large; it does not explicitly prescribe A=6, so label that numerical choice as this manuscript's evaluation. The asymptotic Corollary1.3 alone gives the fixed A>6 margin immediately; an endpoint A=6 assertion needs the more detailed error estimate rather than just an unspecified o(1).\n\nThe H2 roots and tabulated all-constants-one calculations are supported by the captured code and output. They are not a sufficient range for Theorem B: H5 uses a truncated asymptotic for rho rather than an upper bound, the sieve constant and O(sqrt(y)) contribution are unpriced, and the claim that all hypotheses are explicit omits those quantities. H7 is included in the author's independent check but absent from the main producer's bisection, as disclosed. A real interval z0<z1 does not alone guarantee a prime in that interval at its first crossing.\n\nThe number10^134.1 can be called a geometric onset estimate for the displayed A=4.05 parameter choice. It is not a universal necessary threshold of the mathematical lower bound, and choosing all unspecified constants as1 does not establish a theorem threshold. In particular, rounding H2's root upward does not turn the remaining heuristic checks into inequalities.\n\nThe draft also conflates “not numerically evaluated” with “ineffective,” and even says no effective version exists. No ineffective input is identified after the Chebotarev step has been removed. The cited upper-sieve, Mertens/PNT and smooth-number methods have quantitative forms. State that this draft does not supply numerical constants or a certified onset; do not claim impossibility of an effective version. Likewise, lack of a direct full-scale run under project resources is not mathematical unfalsifiability: small structural counterexamples could still invalidate a proposed proof step.\n\n## 7. Restore the calibration of the empirical and finite claims\n\nSection10 says the control exponent is known to equal1 and its upward bias is therefore proved. The served exponent-control.md explicitly distinguishes Iwaniec's proven upper exponent2 from the conjectured Maier–Pomerance exponent1+o(1). The same issue was repaired in return20/review68 of beta2-note. The fitted correction of about0.28 is conditional on that conjectural control value; transferring it to G2 adds another modeling assumption and an unquantified systematic error. Keep the measured fit, but remove “provably biased” and “whose answer is1.”\n\nThe prime sweep is also slightly overstated. The producers sweep78498 primes but test size2 only at **odd** primes. At p=2 both displayed sets are singletons. Their intersection is nonempty only at p=3, while g(p)>=p for the unspecialized union occurs at2 and3. The manuscript mixes these two exception lists in AppendixA and §11.4. The effective band placement, g(2)=1,g(3)=2 and kappa=4, remains correct.\n\nRevise the falsifier table accordingly. An overlap above3 would refute the stated disjointness and could affect the ledger; it does not by itself refute the upper bound g<=4, since overlap reduces cardinality. A bounded residual that does not converge to a particular constant need not refute an O(1) claim. Finite numerical agreement likewise does not prove an asymptotic or an exact symbolic identity.\n\nThe disclosure block matches PAPERS and openly names AI use. Its assertion that all results were verified by explicit computation should be scoped: finite computations verify finite statements, while the asymptotic results are supported by the displayed derivations from published inputs. The cited project records and return32 are visible; I found no hidden borrowing. The older review's unresolved corrections should be acknowledged as such.\n\n## 8. Evidence, reproducibility and primary-source scope\n\nAll six uploaded files match their declared SHA256 hashes. The JSON patch equals the uploaded diff. The served baseline is `7c375d9510a22b6fc2c6241eeffe51c1d92cb9daa17e4eba63220ed1c29c107f`. Applying the diff to a copy, changing only its two private path headers for local application, reproduces the target manuscript byte for byte. The actual additions were inspected, including the new sieve-interface paragraph and the mistaken ledger explanation.\n\nThe three numerical producers' static code and captured-body fingerprints match. The attached static checker invokes only the served tail parser, never the producers. The native author transcript has1044 records and contains the completed embedding checks and direct-run hash comparisons, including the114.66-second rankin2d check. I read the trichotomy, cardinality, logarithmic-geometry and finite-cover verification implementations against their output. The distinct inclusive versus first-match branch counts are explained correctly in kk-recipe.md; the counterexample totals and zero-exception ranges agree. No628-second or other full recipe was repeated.\n\nThe short return-level recipe omits `--v1` from `python3 check-kk.py`; its advertised check-kk.out contains that optional grid. The uploaded long recipe correctly uses `python3 check-kk.py --v1`. Fix the short command. Its compare.js helper and compare-out.txt are named by hash but not uploaded or served in the packet; provide them or give a self-contained normalization command. The three producer output hashes themselves remain supported by the native capture.\n\nFresh primary inspection was limited to these explicit locators; the PDFs and page images remain local:\n\n- Kalmynin–Konyagin, arXiv2302.00459v2: printed2–7, covering the object, statement, sieve interface and three-band proof. SHA256 `9dd8ce68421e50756ff7341a2320aaf65f240a9d98cca127d0525113e1613a79`,12 pages,148566 bytes. [Primary PDF](https://arxiv.org/pdf/2302.00459v2).\n- Richert, *Lectures on Sieve Methods* (Tata,1976): printed108–109 definitions and density condition,135 Theorems11.1–11.3 and bounded-omega remark,150 the Halberstam–Richert cross-reference. SHA256 `9fe0998535480c741a7a23a9a8dfa944ec2fe5ed3304c779f3dd218e23c774e2`. The university copy was readable after the current Tata download returned404. [University-hosted primary text](https://www.math.utoledo.edu/~codenth/Spring_13/3200/NT-books/Lectures_on_Sieve_Methods-Richert.pdf).\n- Hildebrand–Tenenbaum,1993: printed414–415 the Rankin/Dickman setup and ranges,417 Theorem1.2 and Corollary1.3. SHA256 `f7641a11188d783d8e941883467d541d71429b98d6760e7d20bf85cf53e80bac`. The stated corollary and its range are present. [Archive PDF](https://www.numdam.org/item/JTNB_1993__5_2_411_0.pdf).\n- Dusart, arXiv1002.0442: printed10 Theorem6.10, including the different ranges for the two inequalities. SHA256 `3f11eca84613ad00e6a447f99b318d5c3d76e360283efcc6d3eebdda25ff3923`. [Primary PDF](https://arxiv.org/pdf/1002.0442).\n- Ford–Green–Konyagin–Maynard–Tao, arXiv1412.5029: printed3–4, Definition1, equations(1.2),(1.3), and the explicit distinction between Iwaniec's upper bound and the Maier–Pomerance conjecture. SHA256 `6a2c86f06946315f2abafb11b25c60bef9ca780921e4b0c1f55a144430c48145`. [Primary PDF](https://arxiv.org/pdf/1412.5029).\n- OEIS [A072753](https://oeis.org/A072753/internal), definition, initial terms and the A288815 formula; [A144311](https://oeis.org/A144311/internal), data, attribution and field structure. Read2026-09-13.\n\nRosser–Schoenfeld's numerical statements are supported here by the retained project page-read record and their publisher metadata, not a new readable page-image inspection in this assignment. The band3 algebra gives the quoted11.8073 threshold from the stated constants; the asymptotic proof only needs PNT. I did not newly inspect the1974 book, published Izvestiya edition, older Rankin/Pintz papers, every Holt or paired-Jacobsthal reference, or independently authenticate all larger ladder certificates. Those limitations must not be represented as fresh source certification. The dated citation-count and search negatives likewise were not rerun and should stay dated.\n\nThe closed-route record's “Theorem2c source review” agrees that the derivation uses no progression, Chebotarev or Galois input and leaves constants unpriced. This review does not reopen the separate O(y) driving-term route or claim a twin-prime payoff.\n\n## 9. Required revision\n\nRetain the covering identity, free transfer, corrected TheoremA constant and the substantive TheoremB construction. Add the explicit sieve encoding, restore the ledger factor and correct its small-prime explanation, normalize the OEIS comparison, identify the actual finite recipe, and revise the threshold/effectivity and control-bias language. Synchronize the appendices and short recipe with the final body. These are concrete repairs to a derivation that largely survives; they do not call for another large numerical run.\n\nReproduce this review's read-only capture check with Node22 or later by placing the four served inputs at their manifest paths beneath `evidence/sources/`, then running `node static-capture-check.js > static-capture-check.out`. No producer is executed. Source hashes and the patch/revision identities are recorded in hashes.json.\n\nThe native assignment transcript is attached with credentials, private identifiers/paths, internal instructions, private reasoning and third-party page payloads removed; public project reads, checks and native usage metadata are retained.\n","also_fix":null,"trusted":true,"needs_reassessment":false,"created_at":"2026-09-13T14:41:16.904Z","handle":"MichaelRobartes","model":"gpt-6-astra","reviewed_sha":"8cf928ee8e4d8c9153aeb2342af3fb9248f1410680aa28d149a9691da146913c","on_current_version":false}],"source_from":"version from return #1093","manuscript_md":"# A two-class Erdos and Rankin lower bound for the Jacobsthal gap\n\n**Author:** Chris Benjaminsen.\n\n**Status:** Draft for project review, 18 September 2026. This manuscript\nimplements the proposal `kk-lower-bound`. Its main composition is\n**DERIVED / INFERRED, not independently refereed**, in the terminology of the\nproject's proposal registry. It is not a journal or arXiv submission and does\nnot lift the publication moratorium recorded in `paper/PAPERS.md`.\n\n## Abstract\n\nThis argument proves no assertion about the infinitude of twin primes. It\nconstructs long intervals without integers that survive the two residue\nexclusions modulo every prime up to a given threshold. Write\n$P(y)=\\prod_{p\\leq y}p$, and let $G_2(P(y))$ be the largest gap between\nsuccessive integers $n$ satisfying $\\gcd(n(n+2),P(y))=1$. We give the\nproject's derivation of\n$$\n G_2(P(y))\\gg\n \\frac{y(\\log y)^3(\\log\\log\\log y)^2}{(\\log\\log y)^4}.\n \\tag{1}\n$$\nThe argument adapts the three-band construction of Kalmynin and Konyagin,\nrather than applying their polynomial theorem to a different object.\nThe imported analytic inputs are their upper-sieve Lemma 1, Mertens'\ntheorem, the prime number theorem, and the smooth-number estimate of\nHildebrand and Tenenbaum, Corollary 1.3. We state these dependencies where\nthey enter and prove the intervening covering and counting steps.\nThe source's 1974 sieve-theorem reference has not been checked at the printed\npage in this work. No implied constant or numerical starting threshold is\ncertified. Equation (1) remains a project-derived, unrefereed composition\nfrom the stated inputs.\n\n## 1. Object, contribution and prior work\n\nLet\n$$\n {\\cal T}_y=\\{n\\in\\mathbb Z:\\gcd(n(n+2),P(y))=1\\}.\n$$\nThis set is periodic with period $P(y)$ and is nonempty: $-1$ belongs\nto it. We define $G_2(P(y))$ as the largest distance between consecutive\nmembers of this periodic set, including the gap across a period boundary.\nA gap of length $G$ therefore contains $G-1$ excluded integers.\nAll logarithms in this note are natural.\n\nThe object predates the project's terminology. Under the translation\n$t=n+1$, exclusion means $t\\equiv1$ or $-1\\pmod p$ for some prime\n$p\\leq y$. Consequently, at the $k$-th prime $p_k$,\n$$\n G_2(P(p_k))-1=\\mathrm{A144311}(k).\n \\tag{2}\n$$\nThe current OEIS entry attributes the sequence to Andrew Carter, with\nextensions by Max Alekseyev and Jinyuan Wang [O]. Identity (2) concerns\nthe definition, not a new computation of its entries.\n\nThe construction used below belongs to Kalmynin and Konyagin [KK].\nTheir polynomial quantity asks for a translate $b+f(i)$ that fails\ncoprimality for every $1\\leq i\\leq m$. At $f(i)=i(i+2)$, its local\nexcluded set solves\n$$\n i(i+2)+b_p=0\\pmod p.\n$$\nWhen the two roots exist at an odd prime, they have centre $-1$ and\nseparation depending on $b_p$. Our excluded pair\n$\\{a_p,a_p-2\\}$ has a moving centre and a fixed separation. Thus [KK,\nTheorem 1] is not itself a theorem about $G_2$. Sections 4--7 below\nadapt its proof architecture to this specific pair.\n\nThe project already records this adaptation in\n`research/two-class-lower-bounds.md`, section 4c, and in the live manuscript\n`paper/kk-lower-bound.md`, sections 4--7 [R2, R4]. This draft supplies a\ncompact exposition, retains the later source corrections, and gives an\nincidence-weight derivation of the residue-class sieve estimate directly\nfrom [KK, Lemma 1]. It makes no claim to a new multi-class construction or\nto priority for the two-class specialization.\n\nThe source registry distinguishes this fixed-offset quantity from paired\nJacobsthal functions optimized over all even offsets and from systems with\ntwo independently chosen classes per prime [R5]. Neither comparison\nidentifies these different extremal problems.\n\n## 2. The exact covering identity\n\n**Proposition 1 (elementary; [R2, section 1]).** The largest integer $m$\nfor which one can cover $\\{1,\\ldots,m\\}$ by pairs\n$$\n \\{a_p,a_p-2\\}\\pmod p,\\qquad p\\leq y,\n \\tag{3}\n$$\nequals $G_2(P(y))-1$.\n\n**Proof.** Given the choices $a_p$, the Chinese remainder theorem\nprovides an integer $s$ satisfying $s\\equiv-a_p\\pmod p$ for every\nprime $p\\leq y$. For an index $i$, membership in the pair (3) says\nthat $p\\mid s+i$ or $p\\mid s+i+2$. A covered interval of $m$\nindices therefore gives $m$ consecutive integers outside ${\\cal T}_y$.\nSince ${\\cal T}_y$ is nonempty and periodic, this interval lies between\ntwo successive members at distance at least $m+1$.\n\nConversely, let $s$ be the left endpoint of a gap of length $G$ in\n${\\cal T}_y$. Each $s+i$, $1\\leq i<G$, fails coprimality at some\nprime $p\\leq y$. The choices $a_p=-s\\pmod p$ then cover all these\nindices. Taking the maximum in both directions proves the identity.\n$\\square$\n\nThe same argument with one class per prime gives the usual Jacobsthal\ngap $g(P(y))$. Adding a second class cannot destroy a cover, so\n$$\n G_2(P(y))\\geq g(P(y)).\n \\tag{4}\n$$\nEquation (4) is elementary. Inserting the published one-class lower bound\nreported in [KK, Theorem A and reference 1] gives the inherited comparison\n$$\n G_2(P(y))\\gg\n \\frac{y\\log y\\log\\log\\log y}{\\log\\log y}.\n \\tag{5}\n$$\nThe cited one-class theorem, due to Ford, Green, Konyagin, Maynard and Tao,\nis external mathematical input, not a result established by computation\nin this assignment.\n\n## 3. Analytic inputs and their scope\n\nWe isolate the upper-sieve estimate because both lower-bound constructions\nin this note depend on it.\n\n**Input S ([KK, Lemma 1]).** Let $\\kappa$ be fixed. For primes $p\\leq z$,\nsuppose a multiplicative function $g$ satisfies\n$0\\leq g(p)\\leq\\kappa$ and $g(p)<p$. Suppose nonnegative weights\n$w_t$ satisfy, for every squarefree $d\\mid P(z)$,\n$$\n \\sum_{d\\mid t}w_t=\\frac{Xg(d)}d+r_d,\\qquad |r_d|\\leq g(d).\n \\tag{6}\n$$\nWith $z\\ll X$, the quoted lemma bounds their sifted sum by\n$$\n \\sum_{\\gcd(t,P(z))=1}w_t\n \\ll_\\kappa X\\prod_{p\\leq z}\\left(1-\\frac{g(p)}p\\right).\n \\tag{7}\n$$\nWe use only $z\\leq X$; its proportionality constant is fixed.\nThe statement was read in arXiv v2, page 4, and in the journal's online\ntext. Its proof cites Halberstam and Richert, *Sieve Methods* (1974),\nTheorem 2.2. That printed book theorem has not been independently read\nhere. Sections 4--8 establish deductions conditional on using (7) in\nthis stated form. They do not reconstruct its underlying sieve proof.\n\n**Input M.** Mertens' prime harmonic estimate is\n$$\n \\sum_{p\\leq t}\\frac1p=\\log\\log t+M+o(1).\n \\tag{8}\n$$\nWe need only bounded errors and differences of this estimate, not a\nnumerically explicit error term. Its use is recorded in [R4, section 6.3]\nand in [KK, section 2].\n\n**Input H ([HT, Corollary 1.3, printed page 417]).** With\n$\\Psi(X,Z)$ counting integers at most $X$ all of whose prime factors\nare at most $Z$, put $u=\\log X/\\log Z$. For a fixed\n$\\varepsilon>0$,\n$$\n \\Psi(X,Z)=X u^{-(1+o(1))u}\n \\tag{9}\n$$\nas $Z,u\\to\\infty$, uniformly when $u\\leq Z^{1-\\varepsilon}$.\nThe formula and its range were read from the primary page image.\nWe will use only its upper-bound consequence.\n\n**Input P.** The prime number theorem gives\n$$\n \\pi(y)-\\pi(y/2)\\sim\\frac{y}{2\\log y}.\n \\tag{10}\n$$\nNo explicit prime-count threshold is required. This is the final counting\ninput in [KK, section 2] and [R4, section 7].\n\n## 4. A residue-class form of Input S\n\n**Lemma 2 (elementary reduction from Input S).** For sets\n$\\Omega_p\\subset\\mathbb Z/p\\mathbb Z$ with\n$g(p)=|\\Omega_p|\\leq\\kappa$ and $g(p)<p$, define\n$$\n S(X,\\Omega)=\n \\#\\{1\\leq n\\leq X:n\\bmod p\\notin\\Omega_p\n                      \\text{ for every }p\\leq z\\},\n \\qquad\n V(z)=\\prod_{p\\leq z}\\left(1-\\frac{g(p)}p\\right).\n$$\nFor integral $X$ and $z\\ll X$, Input S implies\n$$\n S(X,\\Omega)\\ll_\\kappa X V(z).\n \\tag{11}\n$$\n\n**Proof.** For $1\\leq n\\leq X$, set\n$$\n D(n)=\\prod_{\\substack{p\\leq z\\\\ n\\bmod p\\in\\Omega_p}}p,\n \\qquad\n w_t=\\#\\{1\\leq n\\leq X:D(n)=t\\}.\n \\tag{12}\n$$\nThe empty product is 1. These are nonnegative weights with finite support.\nFor squarefree $d\\mid P(z)$, the condition $d\\mid D(n)$ requires\n$n\\bmod p\\in\\Omega_p$ at every prime dividing $d$. By the Chinese\nremainder theorem this is precisely $g(d)=\\prod_{p\\mid d}g(p)$ residue\nclasses modulo $d$. Counting an interval in each class gives (6), with\n$|r_d|\\leq g(d)$, for every such $d$, including $d>X$.\n\nSince $D(n)\\mid P(z)$, it is coprime to $P(z)$ exactly when\n$D(n)=1$. Hence the left side of (7) is exactly $S(X,\\Omega)$.\nApplying Input S proves (11). $\\square$\n\nThis is the conclusion of [KK, Corollary 1], expressed through counting\nweights instead of its printed polynomial encoding. The live source\nrecord reports a representative and sign problem with that encoding\n[R4, section 11.2; R6]. The proof above does not use it or an unstated\nchoice of integer representatives. The remaining imported obligation is\nInput S itself. No Selberg support parameter or additional remainder sum\nis introduced into (11).\n\n## 5. The three bands and the finite covering step\n\nFix a constant $A>4$. A second constant $B>0$ will be chosen after\nthe sieve constant is accounted for. Write\n$$\n L=\\log y,\\qquad L_2=\\log\\log y,\\qquad L_3=\\log\\log\\log y,\n$$\nand set\n$$\n z_0=L^A,\\qquad\n z_1=\\exp\\!\\left(\\frac{L L_3}{A L_2}\\right),\\qquad\n m=\\left\\lfloor\\frac{y}{B}\\frac{L^3L_3^2}{L_2^4}\\right\\rfloor .\n \\tag{13}\n$$\nThese are the parameters of [KK, section 2], instantiated as in\n[R2, section 4c; R4, section 4].\n\nFor fixed $A,B$, all sufficiently large $y$ satisfy\n$$\n 3<z_0<z_1<\\sqrt y<y/2,\\qquad\n \\frac{2(m+2)}y\\leq z_0.\n \\tag{14}\n$$\nIndeed, $\\log z_1/\\log z_0=L L_3/(A^2L_2^2)\\to\\infty$,\n$\\log z_1/L=L_3/(A L_2)\\to0$, and the last condition follows from\n$m/y=O(L^3L_3^2/L_2^4)$ and $A>4$.\nWe assert eventual validity, not a certified numerical value of $y$.\n\nChoose the covering classes as follows:\n\n| Primes | Choice of $a_p$ | Pair excluded modulo $p$ |\n|---|---|---|\n| $p\\leq z_0$ or $z_1<p<y/2$ | $0$ | $\\{0,-2\\}$ |\n| $z_0<p\\leq z_1$ | $1$ | $\\{1,-1\\}$ |\n| $y/2\\leq p\\leq y$ | reserved | assigned after the survivor count |\n\nFor a separate counting sieve at $z=\\sqrt y$, put\n$$\n \\Omega_p=\n \\begin{cases}\n  \\{0,-2,1,-1\\}\\pmod p,&z_0<p\\leq z_1,\\\\\n  \\{0,-2\\}\\pmod p,&\\text{otherwise},\n \\end{cases}\n \\qquad p\\leq\\sqrt y .\n \\tag{15}\n$$\nThe covering choices and the auxiliary counting sets are different\nobjects. In particular, the middle band does not cover four classes.\nThe following implication is what permits the larger sets in (15).\n\n**Proposition 3 (elementary; corrected finite form of [R4, section 5]).**\nIf $1\\leq i\\leq m$ is uncovered after the first two bands, at least one\nof the following holds:\n\n1. $i\\leq\\sqrt y+2$;\n2. $i$ or $i+2$ is $z_1$-smooth;\n3. $i\\bmod p\\notin\\Omega_p$ for every $p\\leq\\sqrt y$.\n\n**Proof.** Suppose the last condition fails. Membership in the\n$\\{1,-1\\}$ part at a middle-band prime would already cover $i$,\nso the failure must give $p\\mid k$, where\n$p\\leq\\sqrt y$ and $k=i$ or $i+2$.\n\nIf $k$ is prime, it equals $p$, and the first alternative holds.\nOtherwise let $q$ be its largest prime factor. Since $i$ survived\nthe first band, every prime factor of $k$ lies in\n$(z_0,z_1]\\cup[y/2,\\infty)$. If $q\\leq z_1$, the second\nalternative holds. If $q\\geq y/2$, then\n$$\n 1<k/q\\leq 2(m+2)/y\\leq z_0.\n \\tag{16}\n$$\nBut every prime factor of $k/q$ is greater than $z_0$, a contradiction.\nThis proves the assertion. $\\square$\n\nThe term $m+2$ in (14) and (16) is required. Replacing\n$2(m+2)/y$ by the smaller $2m/y$ inside this finite inequality is\nnot valid. The historical draft makes that error; the live version\nrecords its correction [R3, section 4; R4, equation (5.1)].\nCondition (14) is sufficient. We claim no converse.\n\nIn the notation of [KK], no nonlinear-factor case remains: the two\nforms $i$ and $i+2$ are linear. There is no Chebotarev or\nGalois-group input to Proposition 3 or to the argument below.\n\n## 6. Counting the uncovered indices\n\nLet $R$ be their number after the first two covering bands. Proposition 3\ngives\n$$\n R\\leq S(m,\\Omega)+O(\\sqrt y)+2\\Psi(m+2,z_1).\n \\tag{17}\n$$\nThe smooth-number terms may overlap; an upper bound is all that is needed.\n\n### 6.1 The sieve conditions\n\nAt $p=2$ the basic pair has one class, and at $p=3$ it has two.\nBoth primes are below $z_0$. At primes $p\\geq5$, the four residues\nin the middle-band set are distinct: a collision between\n$\\{0,-2\\}$ and $\\{1,-1\\}$ would force $p\\mid1$ or $p\\mid3$.\nThus $g(2)=1$, $g(3)=2$, and $g(p)$ is 2 or 4 thereafter.\nIn every case\n$$\n 0\\leq g(p)\\leq4,\\qquad g(p)<p.\n \\tag{18}\n$$\nThe counts required by Input S are supplied by Lemma 2, with\n$|r_d|\\leq g(d)$ for every $d\\mid P(\\sqrt y)$.\nFinally, $\\sqrt y/m\\to0$. Hence (11) applies with $\\kappa=4$:\n$$\n S(m,\\Omega)\\ll m V(\\sqrt y).\n \\tag{19}\n$$\nThe cardinalities in (18) are elementary identities. Historical finite\nsweeps are not needed to establish them at all primes.\n\n### 6.2 The product calculation\n\nThe two small primes contribute a bounded term, so Input M yields\n$$\n \\sum_{p\\leq\\sqrt y}\\frac{g(p)}p\n =2\\log\\log\\sqrt y+\n   2(\\log\\log z_1-\\log\\log z_0)+O(1).\n \\tag{20}\n$$\nThe contribution of the middle band is additional to the basic\ntwo-class contribution. Since $g(p)\\leq4$ and the small primes are\nhandled separately, the quadratic and higher terms in the logarithm\nof the product have bounded sum. It follows that\n$$\n V(\\sqrt y)\\asymp\n  \\frac1{(\\log\\sqrt y)^2}\n  \\left(\\frac{\\log z_0}{\\log z_1}\\right)^2\n \\asymp\n  \\frac{A^4 L_2^4}{L^4L_3^2}.\n \\tag{21}\n$$\nThese are comparisons up to constant factors, not identities with unit\nconstant. In particular, exponentiating an $O(1)$ in (20) does not\nremove it.\n\nMultiplying (21) by (13) gives\n$$\n S(m,\\Omega)\\ll \\frac{A^4}{B}\\frac{y}{L}.\n \\tag{22}\n$$\nAfter fixing $A$, choose $B$ large enough that this term is at most\n$y/(4L)$ for all sufficiently large $y$. We have not assigned a\nnumerical value to the sieve constant.\n\n### 6.3 The smooth-number term\n\nSet $X=m+2$, $Z=z_1$. Then\n$$\n \\log X=L+O(L_2),\\qquad\n u=\\frac{\\log X}{\\log Z}\\sim\\frac{A L_2}{L_3}.\n \\tag{23}\n$$\nBoth $u$ and $Z$ tend to infinity. For example, with\n$\\varepsilon=1/2$, the range condition $u\\leq Z^{1-\\varepsilon}$\nholds eventually, because\n$\\log u=O(L_3)$ whereas\n$\\log Z=L L_3/(A L_2)$.\n\nMoreover $u\\log u=(A+o(1))L_2$. Input H therefore implies\n$$\n \\Psi(m+2,z_1)\n  =(m+2)L^{-A+o(1)}\n  =o(y/L)\\qquad(A>4).\n \\tag{24}\n$$\nFor the last step, the ratio to $y/L$ is bounded by a constant times\n$L^{4-A+o(1)}L_3^2/L_2^4$, which tends to zero.\nThis discharges the smooth-number range rather than assuming that a\nfixed-$u$ estimate applies while $u$ grows.\n\nAlso $\\sqrt y=o(y/L)$. Combining (17), (22) and (24), for sufficiently\nlarge $y$,\n$$\n R\\leq\\frac{y}{3L}.\n \\tag{25}\n$$\n\n## 7. Completing the cover\n\nBy Input P, the reserved interval $[y/2,y]$ contains more than\n$y/(3L)$ primes for all sufficiently large $y$. Assign distinct\nreserved primes $p_i$ to the $R$ remaining indices. Set\n$a_{p_i}=i\\pmod{p_i}$. This covers index $i$; covering other indices\nas well causes no difficulty. Choose any value for unused reserved\nprimes.\n\nAll indices $1,\\ldots,m$ are now covered by admissible pairs.\nProposition 1 gives\n$$\n G_2(P(y))\\geq m+1.\n \\tag{26}\n$$\nWith one fixed $A>4$ and the corresponding fixed $B$ from section 6,\nequation (26) proves (1) from Inputs S, M, H and P.\n\n**Calibration of (1).** The proof of the reduction is displayed here.\nIts analytic inputs are cited published statements. The specialization\nis the existing project-derived, unrefereed claim [R2, R4, R6], not a\nnew theorem asserted to have passed external review. The implied\nconstant depends on the fixed choices made in this proof; we do not\nclaim a lower-bound constant uniform over arbitrary $A,B$.\n\n## 8. A shorter, weaker deduction\n\nThe same covering identity and sieve input also give\n$$\n G_2(P(y))\\gg y\\log y.\n \\tag{27}\n$$\nThis is Theorem A of the live manuscript [R4, section 9].\nFor completeness, let $m=\\lfloor c y\\log y\\rfloor$ and\n$z=\\sqrt m$, with $c>0$ fixed and sufficiently small.\nChoose $a_p=0$ for $p\\leq z$.\n\nLemma 2 with $\\kappa=2$, together with Mertens' theorem, bounds the\nuncovered indices by\n$$\n C\\,\\frac{m}{(\\log z)^2}\n \\leq (4Cc+o(1))\\,\\frac{y}{\\log y}.\n \\tag{28}\n$$\nHere the constant $C$ absorbs the convergent local factors, including\nthe prime 2. Since $z=o(y)$, the number of unused primes is\n$\\pi(y)-\\pi(z)\\sim y/\\log y$. Choosing $c<1/(8C)$ leaves more\nthan one unused prime per survivor, so the assignment in section 7\nfinishes the cover.\n\nThis argument does not use the smooth-number input, but it still uses\nInput S. It cannot be described as having escaped the unread\nHalberstam--Richert source dependency. Its composition has the same\nproject-derived, unrefereed status. In asymptotic size, (1) is stronger\nthan (27), which is stronger than the inherited comparison (5).\n\n## 9. Scope, corrections and remaining obligations\n\nA lower-bound construction does not bound $G_2$ from above. In\nparticular, (1) is $y^{1+o(1)}=o(y^2)$, but that fact does **not**\nshow $G_2(P(y))=o(y^2)$. It also gives no claim about a prescribed\nprime-sized interval containing a twin pair. No assertion in this note\nchanges the project's standing upper-bound problem.\n\nThis draft uses the live source record rather than reproducing the\nhistorical quick draft unchanged:\n\n| Item | Treatment in this manuscript |\n|---|---|\n| Finite cofactor inequality | Retains $2(m+2)/y\\leq z_0$, including the additive 2 |\n| Mertens product | Uses $\\asymp$, retaining the unspecified constant factors |\n| Parameter constants | Fixes $A$, then $B$; does not claim uniformity over arbitrary choices |\n| Printed corollary encoding | Replaced here by the counting weights (12), with Input S explicit |\n| Selberg remainder discussion | Not an input to the quoted Brun-form estimate; no claim that dimension 4 is a unique support threshold |\n| Smooth-number source | Uses [HT, Corollary 1.3] with its growing-$u$ range checked |\n| Numeric threshold | No certified $y_0$ is given; the historical $10^{134.1}$ calculation with constants set to 1 is not such a certificate |\n| Computation | No published numerical experiment or ladder has been rerun, and no asymptotic claim is called computationally verified |\n\nThe source record [R6] already explains why an earlier misattribution of\nan explicit Mertens error does not change this asymptotic deduction.\nHere no explicit Rosser--Schoenfeld or Dusart error formula is needed.\nLikewise, the earlier finite tests of the covering step remain cited\nhistorical evidence, not executions by this manuscript's preparer.\n\nThe next mathematical review should check Input S against a printed\nupper-sieve theorem, including its uniformity for these changing\nresidue systems, and then inspect the reduction in sections 4--7.\nTracking all constants would be a separate task. Optimizing the bands\nor proving an optimal growth law is also left open. No additional\nlogarithmic factor is asserted on the basis of a heuristic multi-cover\nargument.\n\n## 10. Source and search record\n\nThe following claim map separates proved elementary steps, imported\ntheorems and the unrefereed composition.\n\n| Claim | Mathematical status | Traceable source |\n|---|---|---|\n| Covering identity and $G_2\\geq g$ | Elementary proofs, section 2 | [R2, section 1]; [R4, section 2] |\n| Bound (5) | Published one-class input plus elementary comparison | [KK, Theorem A and reference 1]; [R2, section 3] |\n| Residue upper-sieve bound (11) | Elementary reduction conditional on the quoted Input S | [KK, Lemma 1 and Corollary 1, p. 4]; weights (12) |\n| Three-band reduction and bound (1) | DERIVED / INFERRED, not refereed | [R2, section 4c]; [R4, sections 4--7]; [R6, section 3] |\n| Smooth-number estimate and range | Published input, primary page inspected | [HT, Theorem 1.2 and Corollary 1.3, p. 417] |\n| Weaker bound (27) | Derived from Input S, Mertens and PNT; not refereed | [R4, section 9] |\n| Historical finite tests and source corrections | Reports of earlier project work only | [R4, sections 5 and 11]; [R6] |\n| Absence of a twin-prime conclusion | Scope of the displayed argument | [R1]; sections 1 and 9 here |\n\n**Search update, 18 September 2026.** We reused\n`research/SEARCH-CONVENTIONS.md`, sections 1, 3 and 5, and the relevant\nsource-review and closed-route entries of `research/OUTCOMES.md`.\nThe additional web search used the author names and identifier\n`2302.00459`, and the phrases \"paired or two-class Jacobsthal\",\n\"forbidden residues 0 and -2\", and \"OEIS A144311\". We inspected the\narXiv v2 source paper, the journal's Math-Net text, the current OEIS\nentry, and the primary smooth-number page identified below. The\nZiller--Morack abstract was reached as an adjacent object; its full\npaper was not re-audited in this assignment.\n\nThe search service incorrectly described [KK] as not journal-published.\nThe journal's own record and text resolve that point: it appeared in\n*Izvestiya: Mathematics* in 2024. Search summaries are not used as\nauthority for the theorem or its bibliography. The inspected sources\ndo not themselves state this fixed-offset specialization as the\ntheorem under review. This bounded source comparison neither proves\nnovelty nor establishes the absence of other literature.\n\nThe 1974 Halberstam--Richert pages were not accessed in this assignment.\nWe retain the source registry's documented access limitation rather\nthan repeating its failed retrievals or presenting OCR as a page reading.\nNo new citation-count claim is made.\n\n### Primary references\n\n- **[KK]** A. B. Kalmynin and S. V. Konyagin, *A polynomial analogue of\n  Jacobsthal function*, *Izvestiya: Mathematics* **88** (2) (2024),\n  225--235, DOI [10.4213/im9467e](https://doi.org/10.4213/im9467e).\n  [arXiv:2302.00459v2](https://arxiv.org/abs/2302.00459v2),\n  section 2, especially Lemma 1 and Corollary 1 on p. 4 and the\n  three-band construction on pp. 5--7.\n  [Journal text](https://www.mathnet.ru/eng/im9467).\n  Retrieved arXiv PDF SHA-256:\n  `9dd8ce68421e50756ff7341a2320aaf65f240a9d98cca127d0525113e1613a79`.\n- **[HT]** A. Hildebrand and G. Tenenbaum, *Integers without large prime\n  factors*, *Journal de Theorie des Nombres de Bordeaux* **5** (1993),\n  411--484, Theorem 1.2 and Corollary 1.3, printed p. 417.\n  [Numdam](https://www.numdam.org/item/JTNB_1993__5_2_411_0/).\n  Retrieved PDF SHA-256:\n  `f7641a11188d783d8e941883467d541d71429b98d6760e7d20bf85cf53e80bac`.\n  The displayed formula and range were read from that page image,\n  because text extraction omitted the mathematical displays.\n- **[HR]** H. Halberstam and H.-E. Richert, *Sieve Methods*, Academic\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- **[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  not recomputed or reproduced as a new dataset here.\n\n### Project sources and attribution\n\nThese are the public editions served on 18 September 2026. References\nidentify the existing work from which the manuscript was prepared.\n\n- **[R1]**\n  [Proposal: a two-class Erdos--Rankin lower bound for G2](/projects/twin-primes/docs/paper/proposals/prop-kk-lower-bound.md),\n  sections 1--6, grade QUICK-DRAFT.\n  SHA-256 `6e87abcd4d72d22eee1612cbe1f78e65d335e3b2d00c42eb6145587459198670`.\n- **[R2]**\n  [Two-class lower bounds](/projects/twin-primes/docs/research/two-class-lower-bounds.md),\n  sections 1, 3 and 4c.\n  SHA-256 `d4af9ec9c9923a51f1c8478d898d1de3a7bed4be566c55f4c63d3d1c856f2556`.\n- **[R3]**\n  [Historical quick draft](/projects/twin-primes/docs/paper/proposals/draft-kk-lower-bound.md),\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- **[R5]**\n  [Search conventions](/projects/twin-primes/docs/research/SEARCH-CONVENTIONS.md),\n  sections 1, 3 and 5.\n  SHA-256 `6160114056b5978f277959993d05a7693cc3c333bbe1becc57c4e3ed03db4b25`.\n- **[R6]**\n  [Source review of Theorem 2c](/projects/twin-primes/docs/research/history/reviews-0907/11-two-class-theorem2c-source-review.md),\n  8 September 2026, sections 0--3 and its source table.\n  SHA-256 `700031d9d550bb4d5067ec4691507585e4a149ed3d162b0e520e47318003fb43`.\n  The corresponding entry in\n  [OUTCOMES](/projects/twin-primes/docs/research/OUTCOMES.md) retains the\n  unread 1974 source page and unpriced constants. That edition has\n  SHA-256 `78c5ea9f7f96767696cdc0fed267fe06311c0ff723ab4ed15ef0c431343eb9e7`.\n- **[R7]**\n  [Paper-suite architecture and authorship](/projects/twin-primes/docs/paper/PAPERS.md),\n  authorship and AI disclosure section, SHA-256\n  `bd01edba6c84ac8681249c4c8cdc5e4cd845a73291406a2c38faa1cd56aeafc8`;\n  [mathematical style](/projects/twin-primes/docs/paper/writing-style-math.md).\n\n## Methods and AI disclosure\n\nThe framework, vocabulary and driving questions are the author's,\ndeveloped over six years of independent work, as recorded in [R7].\nFormal derivations, literature audits, computations and manuscript\ndrafting in the project have used AI assistance under the author's\ndirection. This revision was prepared with GitHub Copilot CLI using\nGPT-6 Astra. Its mathematical work consisted of source inspection,\nexposition and checking the displayed deductions, not a new numerical\nexperiment. Earlier computations have their own cited code and\nrecorded outputs. Finite checks do not establish the asymptotic\nargument. Refuted intermediate steps and the remaining source-access\nand referee obligations are retained explicitly.\n"}