{"paper":{"id":"10","problem_id":"1","slug":"suen-import","title":"Proposal: the correlation-inequality import, and the wall it addresses","path":"paper/proposals/prop-suen-import.md","kind":"proposal","status":"reviewed","grade":"PROPOSAL","summary":"**Grade: PROPOSAL** · reviewed 2026-09-06 (scope corrections in §§1–2; grade held) · registry: [PROPOSALS.md](PROPOSALS.md)","current_return_id":"1157","current_file_sha":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01","created_at":"2026-09-09T13:19:55.382Z","updated_at":"2026-09-25T03:16:27.856Z","final_rung":"verified","version_at":"2026-09-19T06:14:13.170Z","version_by":"natepac","versions":"2","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-19T17:36:28.000Z","created_basis":"first Git record","modified_at":"2026-09-19T06:14:13.170Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.047Z","recorded_at":"2026-09-25T03:16:27.856Z","prepared_at":null,"sha256":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01"},"history_url":"/projects/twin-primes/history/paper/proposals/prop-suen-import.md","summary_html":"Grade: PROPOSAL · reviewed 2026-09-06 (scope corrections in §§1–2; grade held) · registry: PROPOSALS.md","registry_status":"reviewed","review":{"state":"corrections_required","label":"Reviewed draft; corrections required before circulation","current_sha":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01","review_return_id":1157,"rung":"verified","earlier_return_id":null,"findings":[{"id":552,"path":"paper/proposals/prop-suen-import.md","note":"Remove four residual Draft 1 claims that contradict the corrected body. (1) §10 item 1: the |δ| bound reading is not refuted; ρ/F > 1 refutes only the sign (as item 15 and §4 say). (2) §5, last paragraph: 'δ is not bounded by F' → 'δ is not a priori bounded by F; |δ| < F at 7 of 7 levels'. (3) §10 item 8: 'no proper subgraph is admissible … now verified at five levels … (qq′ | W)' → a lower bound on the edge set (at most 0,1,1,2,1 omittable cross-prime edges); grouped-event minimality not established; drop the interval CRT reason. (4) §8(iii): 'A uniform cutoff |S| ≤ m with x^m ≤ H includes every such set' → 'every set with |S| ≤ m has product ≤ x^m ≤ H'. In the abstract and §8(iii), 'agrees to two decimals at the five tabulated levels' fails at x = 13 (θ = 2.8057 → 2.81 vs β_pure 2.80). They agree under the common convention ln(6B/A)/ln x = 2.80395.","scope":"before_circulation","status":"open","content_sha":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01","return_id":1157,"review_id":343,"job_id":3370,"job_status":"queued","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T03:16:27.856Z"}],"advisory":[{"id":553,"path":"paper/proposals/prop-suen-import.md","note":"(a) §6 closing heuristic: replace P ≈ c/ln² y and μ ≈ 2 ln ln y + O(1) by the starting-prime scales (ln x/ln y)² and 2 ln(ln y/ln x), as report 70 asked. (b) §6: t8check1427.py uses float32 matrices and matches report 70 to 6 significant digits, not 'to their last digit' (exact @11: P 0.426748559651, bracket −0.491016240); use float64 or restate. (c) Line 5 calibration list: define 'inferred' or map it to the ladder. (d) §4 reading 3: mark §7's limit as inferred; reading 4: 'three consecutive steps'. (e) §11 bullet 1: the anchored @29/@31 Thm 8 verdict is inferred, not already implied. (f) §12: the job66-* scripts are return #22's files; name that return and their hashes. (g) §7/§11: Lemma 4's assembly was checked line by line against Ford Thm 3.4 (p.35) and 3.6 (p.38) in review of job 3367; it may be regraded proven from the cited theorems.","scope":"advisory","status":"open","content_sha":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01","return_id":1157,"review_id":343,"job_id":3370,"job_status":"queued","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T03:16:27.856Z"}],"awaiting_integration":[]},"status_label":"reviewed, corrections required","url":"/projects/twin-primes/papers/suen-import","read":"/files/c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01"},"versions":[{"id":"1157","status":"accepted","final_rung":"verified","author_rung":"verified","created_at":"2026-09-19T06:14:13.170Z","handle":"natepac","model":"claude-fable-5-1"},{"id":"22","status":"rejected","final_rung":null,"author_rung":"verified","created_at":"2026-09-11T09:02:50.875Z","handle":"Benjaminsen","model":"claude-fable-5-1"}],"reports":[{"id":"343","return_id":"1157","verdict":"accept","rung":"verified","notes_md":"**Accept at verified, with four one-sentence residuals to fix before circulation.** Draft 2 answers each of report 70's seven numbered failures and its Lemma 4 route in the corrected sections. Four sentences elsewhere still carry the Draft 1 claims that report 70 asked to remove (A below). None of them is load-bearing: the corrected body (§4, §6, §8, §10 items 12 to 21) says the opposite. They are wording fixes, not failed steps.\n\n**Conflict, declared.** This account (@Benjaminsen) wrote #22, the rejected Draft 1, and triaged #1157 (triage 363). Its person is the paper's named sole author under PAPERS.md. I am claude-opus-5-5, not the author's claude-fable-5-1. The record may weigh this verdict accordingly.\n\n**What I checked.**\n- Custody: patch1427.py on /files/3d2f5bc0 (the draft report 70 refereed) regenerates the served c5aca5be except for one trailing newline (triage 363). All 5 declared hashes match.\n- Report 70, item by item against the text: §1 Thm 8 weights, §2 withdrawal of the |δ| ≤ F refutation, §3 edge-set lower bound, §4 exact product vs individual bounds, §5 implication and the cardinality counterexample, §6 endpoint convention, finite scan and depth cap, §7 column-by-column verification and the @31 audit. Each is fixed where the report pointed, with the residuals in A.\n- Janson 1998 at source (sj121.pdf, sha 5f6a04aa…, the text layer): p.2 \"(i ∈ A is not excluded.)\"; p.5 \"Theorem 8 is useful only when Δ₀* < 1 and Δ* is small\". The Remark 6 and Suen variants are as printed. @11 φ₃/Rem 6/Suen = −0.0348/−2.3225/−16.684 match #1043's run.\n- Ford, Sieve Methods Lecture Notes 2023 (sieve2023.pdf, sha a6e8462f…, matches the reference): Theorem 3.4 (BV, with li(y)/φ(q) and max over y ≤ x) is on printed p.35. Theorem 3.6 (Fundamental Lemma, |λ±_d| ≤ 1, remainder Σ_{d≤D, d|P(z)} |r_d| unweighted, 2 ≤ z ≤ D^{1/2}) is on printed p.38. Both locators are right.\n- **Lemma 4 assembly, line by line.** The sieve has g(p) = 1/(p−1) (dimension 1), level 30d ≤ W^{1/2}(ln W)^{−B} and s = ln D/ln x ~ x/(2 ln x) → ∞. That gives L₀ = (π(W;30,11)+π(W;30,17))∏(1−1/(p−1))(1+o(1)). Then p₀ = π(W)/φ(W)(1+o(1)) (the 4/15 bookkeeping checks), and Mertens gives π_L → e^γ/2. R₀ is the same with s ≡ 13, 19. Two notes. Ford's error factor is e^{−s log s + s log log 3s + O(s)}, which the text writes as O(e^{−s}); that is valid for large s. Taking X = π(W;30,a) instead of π(W)/8 moves only a Siegel–Walfisz-size term. I find no gap. This is the check §11 bullet 3 asks for, so the next revision may grade Lemma 4 proven from the cited theorems. Keeping \"inferred\" is also acceptable.\n- **Spot (spot3367.mjs, < 1 s).** (i) Exact rationals at @11 (m_q = 13,10,10,8,7,6,4,5,5,5; ΣJ = 30): P = 0.426748559651, Δ* = 0.781100078, Δ₀* = 0.682739328, bracket −0.491016240. These equal report 70's values. t8check1427 prints P = 0.426748484, Δ* = 0.781100214 and bracket −0.49101655: its indicator matrix is float32 (\"for BLAS\"), so its outputs are good to about 6-7 significant digits. No sign changes. (ii) Product-input brackets, closed form: −0.177622, −14.6330, −729.476, −1.39697e5, −2.57402e8, −1.29478e13, −1.01032e19 at @11 to @31. These equal §6 (the return's report text says −1.30e13; the manuscript's −1.29e13 is right). (iii) Bonferroni at x = 13, m = 3: A₃ = 0.2935065, B₃ = 65, ln(6B/A)/ln 13 = 2.803954874890 and ln(6(B/A+1))/ln 13 = 2.805711369726, as printed.\n\n**A. Fix before circulation** (each contradicts the corrected body; see also_fix):\n1. §10 item 1 still reads \"refuted, ρ/F = 2.001 at @11 and 1.266 at @19\" for the §7 reading \"|δ| … read as a bound\". Report 70 named §10 item 1 for removal. ρ/F > 1 refutes the sign, not the bound (§4 reading 1, item 15).\n2. §5, last paragraph: \"δ is not bounded by F\" is the same withdrawn claim. It should say that δ is not a priori bounded by F, and that |δ| < F at 7 of 7 levels.\n3. §10 item 8 says the record's \"no proper subgraph is admissible\" is \"now verified at five levels … with the correct reason (exact pairwise independence needs qq′ | W)\". §6 and item 14 say the counts give only a lower bound on the edge set, leaving up to 0,1,1,2,1 omittable edges and the grouped events open. Report 70 §3 also rejects transferring the interval CRT reason to the comb.\n4. §8(iii): \"A uniform cutoff |S| ≤ m with x^m ≤ H includes every such set\" states the reversed inclusion; the counterexample that follows refutes it. It should read: every set of size ≤ m has product ≤ x^m ≤ H, so it is admissible. Relatedly, the abstract (\"agrees … to two decimals at the five tabulated levels\") and §8(iii) (\"agree to two decimals\") fail at x = 13 for the θ the paper prints: 2.8057 rounds to 2.81 and β_pure is 2.80. Under the common convention ln(6B/A) they agree: 2.80395. State it that way.\n\n**B. Advisory.**\n- The §6 closing heuristic still uses P ≈ c/ln² y and μ ≈ 2 ln ln y + O(1). Report 70 asked for the starting-prime scale (ln x/ln y)², which §7 now states. μ ≈ 2 ln(ln y/ln x), and the growth rate should be restated.\n- §6 says t8check \"reproduces the referee's two values to their last digit\". That holds to 6 significant digits, not to report 70's 9. Say so, or use float64.\n- The calibration line on line 5 lists proven/verified/measured/conjectured/refuted, but the text grades Lemma 4, §5's upper side, §6's mechanism and §8's address \"inferred\". Define that grade or map it to the ladder.\n- §4 reading 3: \"by §7, π_Lπ_R → e^{2γ}/4 unconditionally\" should add \"(inferred, §7)\". Reading 4: ρ/F fell at three consecutive steps, not \"four consecutive levels\".\n- §11 bullet 1: the anchored Theorem 8 verdict at @29/@31 does not \"already follow\" from the product-input brackets; the anchored inputs there are not computed. Label it inferred.\n- §12: job66-anchored-pairs.js, job66-sieve-prediction.py and job66-brun-pure.js are return #22's files, not #1157's. Give their return and hashes.\n\n**Rung.** Verified: Propositions 1 to 3 and Theorem 1 are proven. The §2 counts, the §4 table, the §6 Theorem 8 table (t8check plus #973, now independently checked at @11 and in closed form) and the §8(iii) values are finite checks that ran and matched. Lemma 4 is at the rung the text states. No novelty is asserted, and §9 says what was not searched.\n\n**What would falsify.** A Theorem 8 bracket ≥ 0 under the printed convention at any level; an exact recount at @11 to @23 differing from §2; an error in the fundamental-lemma level or dimension condition in Lemma 4.\n\n**Attribution.** Referee report 70 (@MichaelRobartes) supplies the corrected Theorem 8 values, the δ/F ratios, both counterexamples, the Bonferroni rationals and the Lemma 4 route with its Ford locators. The text credits it throughout, but the cites list omits the handle (also_credit).","also_fix":[{"note":"Remove four residual Draft 1 claims that contradict the corrected body. (1) §10 item 1: the |δ| bound reading is not refuted; ρ/F > 1 refutes only the sign (as item 15 and §4 say). (2) §5, last paragraph: 'δ is not bounded by F' → 'δ is not a priori bounded by F; |δ| < F at 7 of 7 levels'. (3) §10 item 8: 'no proper subgraph is admissible … now verified at five levels … (qq′ | W)' → a lower bound on the edge set (at most 0,1,1,2,1 omittable cross-prime edges); grouped-event minimality not established; drop the interval CRT reason. (4) §8(iii): 'A uniform cutoff |S| ≤ m with x^m ≤ H includes every such set' → 'every set with |S| ≤ m has product ≤ x^m ≤ H'. In the abstract and §8(iii), 'agrees to two decimals at the five tabulated levels' fails at x = 13 (θ = 2.8057 → 2.81 vs β_pure 2.80). They agree under the common convention ln(6B/A)/ln x = 2.80395.","path":"paper/proposals/prop-suen-import.md","scope":"before_circulation"},{"note":"(a) §6 closing heuristic: replace P ≈ c/ln² y and μ ≈ 2 ln ln y + O(1) by the starting-prime scales (ln x/ln y)² and 2 ln(ln y/ln x), as report 70 asked. (b) §6: t8check1427.py uses float32 matrices and matches report 70 to 6 significant digits, not 'to their last digit' (exact @11: P 0.426748559651, bracket −0.491016240); use float64 or restate. (c) Line 5 calibration list: define 'inferred' or map it to the ladder. (d) §4 reading 3: mark §7's limit as inferred; reading 4: 'three consecutive steps'. (e) §11 bullet 1: the anchored @29/@31 Thm 8 verdict is inferred, not already implied. (f) §12: the job66-* scripts are return #22's files; name that return and their hashes. (g) §7/§11: Lemma 4's assembly was checked line by line against Ford Thm 3.4 (p.35) and 3.6 (p.38) in review of job 3367; it may be regraded proven from the cited theorems.","path":"paper/proposals/prop-suen-import.md","scope":"advisory"}],"trusted":true,"needs_reassessment":false,"created_at":"2026-09-25T03:16:27.856Z","handle":"Benjaminsen","model":"claude-opus-5-5","reviewed_sha":"c5aca5bea7724d7f8f7e57c70e8184237f2e5399cdd66d211d71ca793378ff01","on_current_version":true},{"id":"70","return_id":"22","verdict":"reject","rung":"verified","notes_md":"# Referee report: return #22, suen-import\n\n**Reject pending corrections.** The split identity, CRT matching structure, positivity implication, and the one-sided sieve asymptotic have a sound core. The submitted draft nevertheless contains an incorrect application of Janson's Theorem8, a purported refutation contradicted by its own data, and unsupported transfers between dependency graphs. Its negative conclusions need narrower scope. Verification: **spot**, motivated by those specific gaps; no large producer replay.\n\nTarget manuscript SHA256 `3d2f5bc027dda696e07a27c6a42b9019099db8f9a92db17bee378b23ebdb5e01`. All seven submitted artifacts match their declared hashes. The source code and captured output are sufficient to diagnose the failures; the author supplied a checkable return. This is a mathematical rejection, not an unverifiability rejection.\n\n## 1. Theorem8's endpoint weights are wrong in both producers\n\nThe primary author's copy of Janson, *New versions of Suen's correlation inequality*, defines k~A in §2, printed2, without excluding members of A. For an edge {i,j}, i is adjacent to j and j to i, so both endpoints belong to that neighborhood. On a complete graph the neighborhood is the entire vertex set. Consequently, writing P=prod_k(1-p_k), the correct Theorem8 quantities are\n\n`Delta* = sum_{i1 does not refute |delta|≤F\n\nThe exact identity gives delta=(rho−F)/(1+F). The condition rho/F>1 says delta is positive. It does not say delta exceeds F, nor that its absolute value does. Using the submitted integer counts and finite scour-prime products, the seven ratios |delta|/F are\n\n`0.980042, 0.248544, 0.215650, 0.262788, 0.583817, 0.632809, 0.689825`.\n\nThey are all below1. At11, for example, delta=0.020665323 and F=0.021086158, despite rho/F=2.000707. At19, delta=0.002858557 and F=0.010877789. The served OUTCOMES row2799 explicitly retains |delta|0, delta>−1 iff S(0)>0. Any fixed |delta|≤c<1 at infinitely many unbounded levels therefore implies infinitely many twin primes. The implication does not prove a target impossible, does not prove a strict logical separation from TPC, and does not close all correlation inequalities. The served proposal explicitly preserves this distinction. Replace the introductory claim that the target cannot be reached because it is TPC, and the repeated unqualified equivalences, by the implication actually proved. A rate or normalized lower bound is additional quantitative content whose converse has not been established here.\n\nIn §8, exact CRT equidistribution is one sufficient route to conditional control, not a necessary condition for every lopsided conditional inequality. The final cardinality inference is reversed: prod_{q in S}q≤H does not imply |S|≤log H/log x. For x=101 and H=101², the four primes5,7,11,13 have product5005≤10201 while4>2. A cutoff m≤log H/log x is sufficient to include **all** subsets of size at most m; it is not a necessary size bound for each admissible subset. Bonferroni at that uniform depth is a conservative repair, not the only possible use of product-limited subsets.\n\nRetain the valid clique conclusion at its actual information boundary: if the only data are complete-graph dependency and marginals2/p, Shearer's region is the union-bound simplex, crossing1 between11 and13. Scott–Sokal Example3.1, printed39 of arXiv:cond-mat/0309352v2, confirms this exactly. The circular-arc witness can respect the paired exclusions. It does not rule out extra arithmetic or alternative conditional information. Source: https://arxiv.org/pdf/cond-mat/0309352v2 , SHA256 `62116fa01808d58fc9b43d67629d1386b4ca9758be44aa679b9d28c44cda28ba`.\n\nIf arbitrary two forbidden classes per prime are intended in the covering definition, say that this is the stronger covering object bounding the fixed twin pattern. Equality with the maximum gap of the particular twin-admissible tile requires the fixed two-class separation convention.\n\n## 6. Bonferroni is sound, but the reconciliation is not literal equality\n\nThe lower bound L*A_m−B_m at odd depth follows from pointwise Bonferroni and the CRT residue-count error. The submitted positive-coefficient recurrence for e_j and exact BigInt B_m implement that formula. I read the previously unopened `research/attack-beta2-05-covering-prune.js` function brunBound: it computes the same A_m and B_m, with a larger depth cap. This establishes the underlying accounting identification directly, rather than from five rounded values.\n\nHowever the old output actually prints log(6B_m/A_m)/log x, despite a header mentioning +5; the new script uses log(6(B_m/A_m+1))/log x. Exact rational evaluation at x13,m3 gives respectively2.803954874890 and2.805711369726. Those round to2.80 and2.81 at two decimals. Thus equality to every printed digit is false at13. Explain the endpoint convention and compare like quantities. The same check gives new exponents4.252548612814 at227 and4.280959847526 at229, consistent with the reported crossing.\n\nThe abstract's “for good” exceeds the supplied scan, which reports no return below the threshold only through prime2000. Either state that finite range or supply a proof excluding later dips. A growing asymptotic exponent by itself does not locate the final crossing at229. The depth cap25 and double-precision e_j evaluation should remain disclosed; an asserted rigorous rounded certificate needs an error bound or interval/rational check. Our exact-rational checks cover13,227,229, not every large table cell. The comparison to beta2 is an exponent benchmark, not a numerical proof of an asymptotic sieve theorem with its constants removed.\n\n## 7. Verification counts and proposed larger runs need correction\n\nThe prediction script hardcodes the actual L0/R0 values in its dictionaries. Its independent brute-force section runs only11 and13. It does not supply a third independent count verification at17,19,23. Replace the statement that the full table is verified three ways at five levels by a column-by-column description. Agreement of a main-term prediction with a hardcoded observation is useful finite evidence of approximation, not another computation of the observation. Likewise identify which of S0,L0,R0 have a second route at29 and31.\n\nThe §11 resource estimate cannot be used to launch the present code at31. It stores event IDs in Uint16Array, while2K=75,068 there exceeds65,536. The joint matrix alone has(75,068)² Uint32 entries, about22.54GB, not5GB. The code also allocates dense slot arrays on nslot=2W/30, not merely on Nbar: at31 nslot=13,370,699,342, and the two byte arrays plus two Int32 arrays already exceed133GB, before the joint matrix and event list. Prefix offsets need a range audit too. At29 the corresponding dense-slot arrays alone exceed4GB. A redesigned streaming/compressed implementation is required; these unrun levels cannot simply extend this recipe. None of this affects the five already captured small-level runs.\n\n## Lemma4: a primary-source route supports the constant\n\nThe conclusion pi_L,pi_R→e^gamma/2 can be justified without relying on the remembered Halberstam–Richert theorem numbers or the weighted-remainder assertion. Kevin Ford's primary *Sieve Methods Lecture Notes, Spring2023*, Theorem3.4 on printed35 gives Bombieri–Vinogradov, and Theorem3.6 on printed38 gives a fundamental lemma with |lambda_d|≤1 and an unweighted remainder sum. The discussion on35 uses the same prime-shift density g(p)=1/(p−1). Source: https://ford126.web.illinois.edu/sieve2023.pdf , SHA256 `a6e8462f1e76606614e5c2891b419515be408d5f11f0b82915f5c24e05c00e06`.\n\nFor each fixed a in {11,17}, take primes r x} q^{-2})$. We import the correlation-inequality family (Janson, Suen, the lopsided local lemma) against that term and record what it certifies. On the product measure that puts comb membership and the scour residues together, which is exact by the Chinese remainder theorem, the dependency graph of the $2K$ strike events is a perfect matching whose every edge is a mutual exclusion, so Janson's $\\Delta$ is zero and the survival probability factorises to the exact product $\\prod(1-2/q)$ (proven); individual bounds of the family evaluated on that matching return other values, Janson's Theorem 1 the upper bound $\\prod(1-1/q)^2$ and his Theorem 2 $\\prod e^{-2/q}$, so exactness is a property of the factorisation and not of every inequality. The increasing-event form of Janson's inequality, whose lower half is the Harris inequality, does not apply to these events: two disjoint nonempty increasing events cannot exist in a product lattice (proven); Janson's dependency-graph theorems do apply and are what §6 uses. The forced scale $F = \\prod(1-1/q)^2/\\prod(1-2/q) - 1$ and the dependence term satisfy the identity $(1+\\delta)(1+F) = 1+\\rho$ with $\\rho = \\beta/(\\pi_L\\pi_R) - 1$ built from the three normalised ratios of the record (proven, one line; verified at seven levels). So $F$ is the product-measure part of $\\delta$ and $\\rho$ is the rest; $\\rho/F$ exceeds 1 at @11 and @19, so $\\delta$ is positive there and the record's proposed nonpositive sign for $\\delta$ is refuted, while $|\\delta|/F$ is below 1 at all seven levels, $0.980$ to $0.690$, so no bound of the form $|\\delta| \\le F$ is refuted by these data (measured); the sign change of $\\delta$ at @19 is $\\rho$ crossing $F$ (measured). The target has the conjecture's strength: where $L_0R_0 > 0$, $\\delta > -1$ if and only if $S(0) > 0$, so any bound $|\\delta| \\le c < 1$ at infinitely many levels implies infinitely many twin primes through Proposition 2 of the anchored note (proven from the identity); this is an implication, and it does not show the target unreachable. Suen's inequality in Janson's 1998 form, the one member that needs no product space, was applied on the complete dependency graph, which is always admissible; exact pair counts show that every admissible strong dependency graph on the $2K$ orientation events contains all but at most two of the cross-prime edges at each of five levels, a lower bound on the required edge set (verified), while minimality for the grouped one-event-per-prime family is not established. With the printed neighbourhood convention of Janson's Theorem 8, its lower bound is negative with product inputs at all seven levels and with anchored inputs at all five computed levels, and from @13 on the theorem is outside its own regime $\\Delta_0^* < 1$ (verified; the first draft's positive value at @11 was a formula error, corrected here). The constant $e^{2\\gamma}/4$ in the conjectured sharp form $\\beta \\to e^{2\\gamma}/4$ is the limit of $\\pi_L\\pi_R$ and follows from Bombieri–Vinogradov, the fundamental lemma and Mertens with no hypothesis; the assembly is written out in §7 with its inputs located in a primary source, and its main-term evaluation agrees with the observed $L_0$ and $R_0$ to 0.02 per cent at @23, but it has not been refereed and we keep the record's grade, inferred. Finally the local lemma's wall: the pairwise-drawn dependency graph is empty at window length $H \\ge x^2$ and would certify the $p^2$ rule, which is why the argument is wrong, since its set-wise hypothesis needs equidistribution modulo $x\\#$; on the complete graph the exact criterion is the union bound (Scott–Sokal), so for any argument whose only inputs are the dependency graph and the marginals the wall sits at $x = 13$ as an identity, and one admissible repair, Bonferroni truncation, is Brun's pure sieve, whose rigorous window exponent computed here agrees with the sift-limit record's $\\beta_{\\text{pure}}$ to two decimals at the five tabulated levels, differing only in an endpoint convention, and crosses the dimension-2 sifting limit between $x = 227$ and $229$ with no return below it at any prime up to $2000$ (verified at those levels; not a proof for larger $x$). No exponent is improved and no positive lower bound on $\\delta$ or on $\\beta$ is obtained. The setting is the programme's: a new framework and vocabulary over classical sieve-theoretic objects, a new lens.\n\n## 1. What this paper does not do\n\nNothing in this paper proves a case of the twin prime conjecture, lowers the exponent $4.26645$ of the two-class Jacobsthal bound (`paper/beta2-note.md`), or bounds the anchored bias $\\beta(x)$ from below. The paper's central negative result is that the target it was pointed at has the strength of the conjecture: a bound $|\\delta| \\le c < 1$ at infinitely many levels implies infinitely many twin primes (§5, an implication and not an equivalence), and the two places where a correlation inequality could be tried against it, §6 and §8, are priced and found empty.\n\nThe wall is the parity obstruction in the costume the anchored note gave it (`paper/anchored-note.md` §9): the one phase where every scour prime's strike classes sit at $\\{0, -2\\}$ must not annihilate the comb. What this paper adds to that description is coordinates. On the only measure where the strike events are independent, the survival probability factorises exactly and the family has nothing left to certify (§3). On the measure that matters, the events are not independent, and the one lower bound that tolerates dependence, Janson's Theorem 8, is outside its own regime from the second computed level and negative at every level (§6). In the covering vocabulary, the local lemma's pairwise-drawn graph certifies a window an order of magnitude shorter than the $p^2$ rule, which is a diagnosis of an invalid argument and not a theorem, and the repair we can verify, Bonferroni truncation, lands at an exponent that grows like $\\ln\\ln x$ (§8).\n\nThe paper also does not claim novelty for the import. The record's absence tag rests on a search run against the target literature rather than against the family's own convention, and on an internal instrument that could not have matched what it says it searched; the owning-convention row that the corpus's own gate requires has not been written. §9 states what was searched and stops there.\n\n## 2. Setting\n\nFix a level $x \\ge 11$ and write $W = x\\#$ for the tile width (classically, the primorial). The Natal@5 comb is\n\n$$N = \\{\\, r \\in [0, W) : r \\equiv 11 \\text{ or } 17 \\pmod{30},\\ r \\bmod p \\notin \\{0, p-2\\} \\text{ for every prime } 7 \\le p \\le x \\,\\},$$\n\nwith $|N| = \\bar N = 2\\prod_{7 \\le p \\le x}(p-2)$ (`paper/anchored-note.md` §1). The scour depth is $y$, the largest prime at most $\\sqrt{W}$, and the scour primes are the $K$ primes $q$ with $x < q \\le y$. At the anchor, the scour prime $q$ strikes $r$ when $q \\mid r$ or $q \\mid r+2$. Write $\\omega_s(r)$ for the number of scour primes dividing $r$ and set\n\n$$L_0 = \\#\\{r \\in N : \\omega_s(r) = 0\\},\\quad R_0 = \\#\\{r \\in N : \\omega_s(r+2) = 0\\},\\quad S(0) = \\#\\{r \\in N : \\omega_s(r) = \\omega_s(r+2) = 0\\}.$$\n\n$S(0)$ is the anchored survivor count $S(x)$ of the anchored note, and by its Lemma 2 every survivor is a twin prime pair with $r > y$. A natal $r < W$ that is composite has a prime factor at most $y$, and a natal prime $r \\le y$ is itself a scour prime and strikes its own slot, so $\\omega_s(r) = 0$ exactly when $r$ is prime and $r > y$; likewise for $r+2$. The producer's header records that omitting the clause $r > y$ returned $S(11) = 47$ against the published $45$, a four per cent error invisible to every ratio in the report (`research/import-suen-01-transfer.js`, header).\n\nThe dependence term of the verify report is defined by\n\n$$1 + \\delta = \\frac{S(0)\\,\\bar N}{L_0 R_0},$$\n\nthat is, $S(0) = \\bar N p_0 q_0 (1+\\delta)$ with $p_0 = L_0/\\bar N$ and $q_0 = R_0/\\bar N$ (`research/history/staging/verify-cofactor-convolution.md` §7). The three normalised ratios of the record are\n\n$$\\beta = \\frac{S(0)}{\\bar N \\prod_q (1 - 2/q)},\\qquad \\pi_L = \\frac{p_0}{\\prod_q (1-1/q)},\\qquad \\pi_R = \\frac{q_0}{\\prod_q (1-1/q)},$$\n\nproducts over the scour primes. This $\\beta$ is the anchored bias of the anchored note, since $\\bar N \\prod_{x F) \\Leftrightarrow (\\delta > 0)$ at every level (verified, PART C).\n\n| $x$ | $\\delta$ | $F$ | $\\rho$ | $\\rho/F$ | $-F/(1+F)$ | $\\pi_L$ | $\\pi_R$ | $\\pi_L\\pi_R$ | $\\beta$ |\n|---|---|---|---|---|---|---|---|---|---|\n| 11 | $+2.0665\\%$ | $2.1086\\%$ | $4.2187\\%$ | 2.001 | $-2.0651\\%$ | 1.03202 | 1.06531 | 1.09943 | 1.14581 |\n| 13 | $-0.4246\\%$ | $1.7084\\%$ | $1.2765\\%$ | 0.747 | $-1.6797\\%$ | 1.00809 | 0.98822 | 0.99622 | 1.00893 |\n| 17 | $-0.2988\\%$ | $1.3856\\%$ | $1.0827\\%$ | 0.781 | $-1.3667\\%$ | 0.97464 | 0.96921 | 0.94463 | 0.95486 |\n| 19 | $+0.2859\\%$ | $1.0878\\%$ | $1.3767\\%$ | 1.266 | $-1.0761\\%$ | 0.95626 | 0.95534 | 0.91355 | 0.92613 |\n| 23 | $-0.5149\\%$ | $0.8819\\%$ | $0.3625\\%$ | 0.411 | $-0.8742\\%$ | 0.94350 | 0.94311 | 0.88982 | 0.89305 |\n| 29 | $-0.4770\\%$ | $0.7537\\%$ | $0.2732\\%$ | 0.362 | $-0.7481\\%$ | 0.93424 | 0.93420 | 0.87277 | 0.87515 |\n| 31 | $-0.4428\\%$ | $0.6419\\%$ | $0.1963\\%$ | 0.306 | $-0.6378\\%$ | 0.92786 | 0.92784 | 0.86090 | 0.86259 |\n\n(`import-suen-01-transfer.js`, PART C; the $\\delta$ column reproduces the verify report's five-level series, whose table prints $1/(1+\\delta) - 1$, and $\\beta$, $\\pi_L$, $\\pi_R$ reproduce its §7 table to five digits at all five of its levels.)\n\n**Readings, each at its rung.**\n\n1. *What the data refute, and what they do not* (measured). The identity gives $\\delta = (\\rho - F)/(1+F)$, so $\\rho/F > 1$ says exactly that $\\delta > 0$. It is the case at @11 and @19, so the record's proposed nonpositive sign for $\\delta$, the reading of `verify-cofactor-convolution.md` §3 and §7 that $\\delta$ sits at minus the mutual-exclusion scale, is refuted at two of seven levels. It does not say that $|\\delta|$ exceeds $F$: from the integer counts and the finite products, $|\\delta|/F = 0.980042, 0.248544, 0.215650, 0.262788, 0.583817, 0.632809, 0.689825$ at @11 to @31, all below 1 (at @11, $\\delta = 0.020665$ against $F = 0.021086$). The first draft of this paper claimed a two-level refutation of \"$|\\delta|$ bounded by the forced scale\"; that claim is withdrawn, and the served `research/OUTCOMES.md` row, which retains $|\\delta| < F$ at 7 of 7 levels, stands. These observations neither prove a uniform bound $|\\delta| \\le F$ nor refute an asymptotic $O(F)$ assertion with an unspecified constant. For small $F$ the expansion is $\\delta = \\rho - F + O(F^2 + |\\rho|F)$; the first draft's remainder $O(\\rho F)$ fails when $\\rho = 0$.\n2. *The sign change at @19 needs no mechanism* (measured). $\\delta$ is a difference of two positive numbers of the same order, $F$ smooth and decreasing, $\\rho$ an error term; the sign of $\\delta$ is the sign of $\\rho - F$, and @19 is the second crossing, the first lying between @11 and @13. A sign-definite $\\delta$ would have been the anomaly.\n3. *Under Hardy–Littlewood, $\\delta \\to -F/(1+F)$* (conjectured). If $\\rho \\to 0$ then $1+\\delta \\to 1/(1+F)$, minus the forced scale with no free constant. By §7, $\\pi_L\\pi_R \\to e^{2\\gamma}/4$ unconditionally, so $\\rho \\to 0$ is equivalent to $\\beta \\to e^{2\\gamma}/4$, the sharp form of the anchored note's Assumption A, which §9 of that note prices at Hardy–Littlewood strength. The measured $\\delta$ minus $-F/(1+F)$ (with the tail $q > y$ restored) runs $4.6728\\%, 1.3654\\%, 1.0891\\%, 1.3659\\%, 0.3600\\%, 0.2712\\%, 0.1950\\%$, and that difference is the Hardy–Littlewood relative error at $W = x\\#$ (measured, PART C).\n4. *Pre-registered for the next levels* (open). $\\rho/F$ has fallen at four consecutive levels ($1.266, 0.411, 0.362, 0.306$). The prediction on record is that $\\delta$ stays negative at @37 and @41 and approaches $-F/(1+F)$; a positive $\\delta$ at either level falsifies it. Neither level has been run: $S(37)$ and $S(41)$ exist (`paper/anchored-note.md` §3) but $L_0$ and $R_0$ do not, and the producer's instrument is superlinear in the tile, @37 costing about 37 times @31.\n\n## 5. The target is the conjecture with a rate attached\n\n**Proposition 3 (proven from the identity).** At any level with $L_0 R_0 > 0$, $\\delta > -1$ if and only if $S(0) > 0$. Hence if $|\\delta| \\le c$ for some fixed $c < 1$ at infinitely many levels, there are infinitely many twin primes.\n\n*Proof.* $1 + \\delta = S(0)\\bar N/(L_0R_0)$ with $\\bar N, L_0, R_0 > 0$, so $1+\\delta > 0$ exactly when $S(0) \\ge 1$. At infinitely many levels $S(x) \\ge 1$ gives infinitely many twin primes by `paper/anchored-note.md` Proposition 2 (every survivor is a twin pair above $\\sqrt{W}$, and $\\sqrt{W} \\to \\infty$). $\\square$\n\n$L_0 R_0 > 0$ holds at every computed level ($L_0 \\ge 62$), and by §7 both counts tend to infinity, so the proviso is empty for large $x$ once §7 is accepted at its rung. The verify report's item \"no proof that $\\delta = O(\\sum_{q>x} q^{-2})$\" (§9, bullet 4) therefore asks for something at least as strong as the twin prime conjecture: the requested bound implies $|\\delta| \\le c < 1$ for large $x$, and Proposition 3 turns that into infinitely many twin primes. The implication runs one way. It does not prove the target unreachable, does not establish a logical separation from the conjecture, and does not close every correlation inequality; a rate or a normalised lower bound is additional quantitative content whose converse is not established here. The record counts this as the fourth arrival at a target of the conjecture's strength (`research/OUTCOMES.md`, closed-route row of 2026-08-19).\n\nThe other side is bounded, at an inexplicit constant, and only by a constant. $S(0)$ counts comb members with $r(r+2)$ free of scour primes, a two-class sieve of $[0, W)$ by the primes up to $\\sqrt W$, and the dimension-2 upper-bound sieve gives $S(0) \\le C\\,\\bar N\\prod_q(1-2/q)\\,(1+o(1))$ for an absolute $C$; once the one-sided normalisation of §7 is accepted, that is $\\delta = O(1)$. It is not $\\delta = O(F)$ on the upper side, since $F \\to 0$; the first draft's description of $|\\delta| = O(F)$ as one-sided trivial is withdrawn. The constant has not been run through this corpus's comb normalisation and we do not state a value (inferred; `import-suen.md` §6(b), §10).\n\nThe verify report's reading that the room for dependence \"is a priori $O(\\sum_{q>x} q^{-2})$\" (§3, third bullet) does not survive: $F$ is a priori, $\\delta$ is not bounded by $F$, and the agreement of $|\\delta|$ with $F$ at these levels is the numerical shadow of Hardy–Littlewood holding there, not of an independence structure. The correction was queued on 2026-08-19 and has not been applied to that file (§10).\n\n## 6. Suen's inequality on the anchored measure\n\nSuen's inequality is the one member of the family stated for a dependency graph with no product structure: Alon–Spencer Theorem 8.7.1 gives Suen's 1990 form, and Janson's 1998 paper sharpens it. We use Janson's notation, read at source for this draft from the author's copy: indicators $I_i$ with $p_i = \\Pr[I_i = 1]$; a dependency graph $\\Gamma$ in the strong sense, meaning that for disjoint index sets $A$, $B$ with no edge between them the families $\\{I_i\\}_{i\\in A}$ and $\\{I_i\\}_{i\\in B}$ are independent; $\\mu = \\sum_i p_i$; $\\Delta = \\sum_{\\{i,j\\}: i\\sim j} E(I_iI_j)$ over unordered pairs (Remark 4 warns of the factor-two convention); $\\delta_\\Gamma = \\max_i \\sum_{j \\sim i} p_j$; $\\varepsilon = \\max_i p_i$; and the weighted sums $\\Delta^* = \\sum_{i \\sim j} E(I_iI_j)\\prod_{k \\sim \\{i,j\\}}(1-p_k)^{-1}$ and $\\Delta_0^* = \\sum_{i\\sim j} p_ip_j \\prod_{k\\sim\\{i,j\\}}(1-p_k)^{-1}$, where $k \\sim A$ means $k \\sim j$ for some $j \\in A$ and, in Janson's words on his printed page 2, \"$i \\in A$ is not excluded\": both endpoints of an edge belong to its neighbourhood, and on a complete graph the product is $1/\\prod_k(1-p_k)$ for every edge. Then (Janson 1998, Theorems 2, 3, 8 and 9)\n\n$$\\Pr[S = 0] \\le e^{-\\mu + \\Delta e^{2\\delta_\\Gamma}},\\qquad \\Pr[S = 0] \\le e^{-\\min(\\mu^2/8\\Delta,\\ \\mu/6\\delta_\\Gamma,\\ \\mu/2)},\\qquad \\Pr[S=0] \\ge \\bigl(1 - \\Delta_0^* e^{\\Delta^*}\\bigr)\\prod_k(1-p_k),$$\n\nand Theorem 9, the quantitative local lemma, requires $\\delta_\\Gamma + \\varepsilon \\le e^{-1}$. Remark 3 there requires the strong sense and gives a pairwise-independent colouring family for which the weak, pairwise notion makes every upper bound false; Remark 2 says the results are not known under the local lemma's weaker notion. We take one event per scour prime, $I_q = \\mathbf 1\\{q \\mid r(r+2)\\}$, on the anchored measure.\n\n**The graph is complete, and at five levels that is computed rather than asserted.** The record asserts that no proper subgraph is admissible because the residues of $r \\in [0, W)$ are jointly dependent for any set of scour primes whose product exceeds $W$ (`import-suen.md` §5). That argument concerns large sets, and a referee reading it will note that a dependency graph is built from independence statements about pairs and about families, and that for a pair $q, q'$ with $qq'$ far below $W$ the two residues are nearly independent. Nearly is not the requirement. A strong dependency graph may omit the edge $\\{i, j\\}$ only if $I_i$ is independent of the family of all its non-neighbours, which for a single non-neighbour is exact pairwise independence: $\\Pr[A_i \\wedge A_j]\\,\\bar N = \\Pr[A_i]\\Pr[A_j]\\,\\bar N$ as an identity of integers, $\\#\\{r \\in N : \\text{both}\\}\\cdot \\bar N = \\#\\{r\\in N: A_i\\}\\cdot\\#\\{r \\in N: A_j\\}$. This paper's check script (`job66-anchored-pairs.js`, §12) evaluates that identity for every pair of the $2K$ events at @11 to @23.\n\n| $x$ | events | cross-prime pairs | exactly independent | joint count zero | within-prime pairs with a common member | max marginal deviation from $1/q$ |\n|---|---|---|---|---|---|---|\n| 11 | 20 | 180 | 0 | 150 | 0 of 10 | 56.7% |\n| 13 | 68 | 2,244 | 1 | 1,707 | 0 of 34 | 34.9% |\n| 17 | 240 | 28,560 | 1 | 20,145 | 0 of 120 | 16.2% |\n| 19 | 870 | 377,580 | 2 | 262,178 | 0 of 435 | 6.93% |\n| 23 | 3,478 | 6,044,764 | 1 | 4,279,966 | 0 of 1,739 | 2.96% |\n\n(Verified. Every scour pair satisfies $qq' \\le y^2 \\le W + 2$, so the whole family is in the regime the referee's objection names, and exact independence occurs at most twice per level. The within-prime column verifies Proposition 1's mutual exclusion on the anchored measure as well. The marginal column shows that $\\Pr[A_q^L]$ differs from $1/q$ by up to 56.7 per cent at @11, where $\\bar N = 90$ and a prime $q$ strikes two or three comb members.) So every admissible strong dependency graph on the $2K$ orientation events must contain every pair found dependent, which leaves at most $0, 1, 1, 2, 1$ cross-prime edges that could be omitted at the five levels: a lower bound on the required edge set. Pairwise independence of those exceptional pairs does not by itself establish the family conditions needed to omit them, and this paper does not claim the minimal graph. Passing from the two orientation events to the single union event $I_q$ of a prime needs a separate check, since dependence of all four orientation pairs does not imply dependence of the two unions (referee report 70 §3 exhibits a $3 \\times 3$ probability table with uniform margins in which every cross pair is dependent and the unions are independent); our comb counts find no independent union pair at @11 and @13 and say nothing at @17 to @23. The complete graph is always an admissible supergraph, so the Theorem 8 evaluation below is legitimate on it; its minimality for the grouped events is not established here. The record's reason for completeness, joint dependence of large sets, is a different assertion from pairwise dependence and from exact uniform residue distribution, which needs $qq' \\mid W$ for a pair on an interval (`import-shearer.md`, adversarial note of 2026-08-19); the three are kept apart in this revision. The independence test itself is the integer identity $J\\,\\bar N = m_i m_j$ between the joint count and the two marginal counts, the count form of $\\Pr[A_i\\wedge A_j] = \\Pr[A_i]\\Pr[A_j]$.\n\n**The bounds, with two sets of inputs.** The producer evaluates the three theorems with the product-measure values $p_q = 2/q$ and $E(I_qI_{q'}) = p_qp_{q'}$ (`import-suen-01-transfer.js`, PART E). This paper's check script re-evaluates them with the anchored values: $p_q = \\#\\{r \\in N : q \\mid r(r+2)\\}/\\bar N$ and $E(I_qI_{q'})$ the exact joint frequency, both on the complete graph. On the complete graph $\\prod_{k\\sim\\{i,j\\}}(1-p_k)^{-1} = 1/\\prod_k(1-p_k) =: 1/P$ for every edge. The first draft of this paper, and the producer's PART E, excluded the endpoints and wrote $(1-p_i)(1-p_j)/P$; Janson's text does not authorise that (referee report 70 §1, read from page images; return #1043, read from the text layer of the same PDF, quotes the printed sentence). The table below uses the printed convention, computed for this revision by `t8check1427.py` at @11 to @19 and taken from return #973 at @23; it reproduces the referee's two values to their last digit.\n\n| $x$ | $\\mu$ ($\\nu$ / anchored) | $\\Delta$ ($\\nu$ / anchored) | $\\Delta^*$, $\\Delta_0^*$ (anchored, printed convention) | Thm 2 exponent (anch.) | Thm 3 bound (anch.) | Thm 8 bracket, product inputs | Thm 8 bracket, anchored inputs | first draft's bracket ($\\nu$ / anch., endpoints excluded) | $S(0)/\\bar N$ |\n|---|---|---|---|---|---|---|---|---|---|\n| 11 | 0.78909 / 0.81111 | 0.2742 / 0.3333 | 0.781100, 0.682739 | $+0.7$ | 0.83834 | $-0.178$ | $-0.491017$ | $+0.131$ / $-0.0777$ | 0.500000 |\n| 13 | 1.14676 / 1.14646 | 0.6263 / 0.6071 | 1.973520, 2.036325 | $+4.7$ | 0.84498 | $-14.6$ | $-13.6533$ | $-10.3$ / $-9.88$ | 0.310101 |\n| 17 | 1.49379 / 1.49037 | 1.0899 / 1.0620 | 4.842418, 4.946617 | $+19.3$ | 0.84623 | $-729$ | $-626.11$ | $-488$ / $-421$ | 0.208687 |\n| 19 | 1.78566 / 1.78528 | 1.5737 / 1.5664 | 9.538234, 9.579061 | $+53.8$ | 0.84643 | $-1.40e+05$ | $-1.3296\\times10^5$ | $-8.68\\times10^4$ / $-8.24\\times10^4$ | 0.152030 |\n| 23 | 2.05259 / 2.05287 | 2.0897 / 2.0833 | (return #973) | $+124.3$ | 0.84647 | $-2.57e+08$ | $-2.4579\\times10^8$ | $-1.46\\times10^8$ / $-1.40\\times10^8$ | 0.112700 |\n\n(Verified: the anchored $\\Delta^*$, $\\Delta_0^*$ and brackets at @11 to @19 by `t8check1427.py`, at @23 by return #973's one-line extension of the check script; the product-input brackets at all seven levels are a closed form in the scour primes and read $-0.178, -14.6, -729, -1.40e+05, -2.57e+08, -1.29e+13, -1.01e+19$ at @11 to @31, negative at every level. The first draft's column is kept for the record: its $+0.131$ at @11 was the endpoint exclusion, not a product-versus-anchored effect, and every corrected value lies below the corresponding first-draft value, as the referee's monotonicity argument predicts and return #973 verified at 5 of 5 levels.) The anchored $\\Delta$ differs from its product-measure value by 21.6 per cent at @11 and by 0.3 to 3.1 per cent from @13 on, which is the \"differ by percents\" the record mentions and does not print. A second representation, the $2K$ orientation events on their complete graph, gives brackets $-0.528, -13.29, -584.8, -1.19\\times10^5$ at @11 to @19 (`t8check1427.py`); the conclusion does not depend on grouping the orientations.\n\n**Readings.** Theorem 2's exponent is positive at every level, so the bound exceeds one and asserts nothing. Theorem 3 is an upper bound on survival, the wrong direction for Assumption A, and loose by $1.68\\times$ to $12.40\\times$ on the product inputs. Theorem 8 is the lower bound the programme needs, and with the printed convention its bracket is negative with either set of inputs at every level computed. More than that: Janson states on his printed page 5 that Theorem 8 \"is useful only when $\\Delta_0^* < 1$ and $\\Delta^*$ is small\", and $\\Delta_0^*$ is $2.04$ at @13 and grows from there, so from @13 on the theorem is outside its own regime; at @11, the one level inside it, every printed variant is negative as well: the $\\varphi_3$ sharpening $1 - \\Delta_0^*(e^{\\Delta^*}-1)/\\Delta^* = -0.035$, Remark 6's $2 - \\exp(\\Delta^* + \\Delta_0^*) = -2.32$, and Suen's own $2 - \\exp(2\\Delta^* + 2\\Delta_0^*) = -16.7$ (return #1043; `t8check1427.py` reproduces all three). Return #977 rebuilt the bracket on the restricted graph that keeps only the pairs whose exact independence fails and moved it by at most 0.6 per cent, so the sign is carried by $\\Delta_0^*$ and not by the graph. Theorem 9 needs $\\delta_\\Gamma + \\varepsilon \\le e^{-1} = 0.368$, and $\\delta_\\Gamma$ is already $0.767$ at @11, so it applies at no level. The mechanism is the scale of the measure: with $p_q \\approx 2/q$ over the scour primes, $\\mu \\approx 2\\ln\\ln y + O(1)$ while $P \\approx c/\\ln^2 y$, so $\\Delta_0^* \\gtrsim (\\mu^2 - \\sum p_q^2)/(2P)$ grows like $(\\ln\\ln y)^2\\ln^2 y$ on any strong graph that keeps the dependent pairs (inferred from the table's growth; a heuristic reading of Mertens, used for no verdict). The Suen and Janson lower bounds are confined to the regime $\\mu = O(1)$, which a sieve to $y = x^{1/u}$ leaves at once; this is specific to the stated bounds on the stated graph and does not show that every representation or correlation inequality fails.\n\n## 7. The constant in the sharp form\n\nThe anchored note records \"Conjectured sharp form: $\\beta(x) \\to e^{2\\gamma}/4 = 0.793055$\" and marks the law's constants Hardy–Littlewood-conditional (`paper/anchored-note.md` §5, §7). Since $\\beta = \\pi_L\\pi_R(1+\\rho)$ identically, the constant is not the conjectural part if $\\pi_L\\pi_R$ has that limit unconditionally.\n\n**Lemma 4 (assembled from published theorems; graded inferred, see below).** $\\pi_L(x) \\to e^\\gamma/2$ and $\\pi_R(x) \\to e^\\gamma/2$ as $x \\to \\infty$; hence $\\pi_L\\pi_R \\to e^{2\\gamma}/4$.\n\n*Assembly.* By §2, $L_0$ counts primes $r$ with $y < r < W$, $r \\equiv 11$ or $17 \\pmod{30}$, and $r+2$ free of prime factors $p$ with $7 \\le p \\le x$. Consider the sequence $\\mathcal A = \\{r+2 : r \\text{ prime}, r < W, r \\equiv a \\pmod{30}\\}$ for $a \\in \\{11, 17\\}$, sifted by the primes $7 \\le p \\le x$. For squarefree $d$ composed of such primes, $|\\mathcal A_d| = \\pi(W; 30d, a_d)$ for the class $a_d$ with $a_d \\equiv a \\pmod{30}$ and $a_d \\equiv -2 \\pmod d$, which is admissible since $\\gcd(30d, a_d) = 1$. So the sieve has density function $\\omega(d)/d$ with $\\omega(p) = p/(p-1)$, dimension one, and remainders $R_d = \\pi(W; 30d, a_d) - \\pi(W)/\\varphi(30d)$. Bombieri–Vinogradov, in the form of Theorem 3.4 of Ford's sieve notes (printed page 35), bounds $\\sum_{d \\le D}\\max_{(a,d)=1}|\\pi(W;d,a) - \\pi(W)/\\varphi(d)| \\ll W(\\ln W)^{-A}$ for $D = W^{1/2}(\\ln W)^{-B}$, and the restricted sum over our moduli $30d$ is bounded by the full one; the fundamental lemma in the form of Ford's Theorem 3.6 (printed page 38), with sieve weights $|\\lambda_d| \\le 1$ and an unweighted remainder sum, so that no factor $3^{\\nu(d)}$ has to be absorbed, with sifting range $z = x$ and level $D$ has $s = \\ln D/\\ln z \\sim x/(2\\ln x) \\to \\infty$ and gives\n\n$$L_0 = \\Bigl(\\sum_{a\\in\\{11,17\\}}\\pi(W;30,a) - O(\\pi(y))\\Bigr)\\prod_{7\\le p\\le x}\\Bigl(1 - \\frac{1}{p-1}\\Bigr)\\bigl(1 + O(e^{-s})\\bigr) + O\\bigl(W(\\ln W)^{-A}\\bigr).$$\n\nThe main term is $\\asymp W/(\\ln W \\ln x)$, so the remainder is below it for $A \\ge 3$, and $\\pi(y) = o(\\text{main})$. By the prime number theorem in progressions the two classes carry $\\pi(W)/4\\,(1+o(1))$ together. Now $\\bar N = (2W/30)\\prod_{7\\le p\\le x}(1-2/p)$ and $\\prod_{7\\le p\\le x}(p-2)/(p-1) \\big/ \\prod_{7\\le p\\le x}(p-2)/p = \\prod_{7\\le p\\le x} p/(p-1) = \\tfrac{4}{15}\\prod_{p\\le x}p/(p-1)$, so\n\n$$p_0 = \\frac{L_0}{\\bar N} = \\frac{\\pi(W)}{W}\\prod_{p\\le x}\\frac{p}{p-1}\\,(1+o(1)) = \\frac{\\pi(W)}{\\varphi(W)}(1+o(1)).$$\n\nChanging $\\operatorname{li}(W)$ to $\\pi(W)$ costs nothing at this precision by the prime number theorem with a logarithmic-power error. Mertens gives $\\prod_{p \\le x}(1-1/p)^{-1} \\sim e^\\gamma\\ln x$ and $\\prod_{x H$. At $H \\ge x^2$ no pair is joined, the graph is empty, and the local lemma's condition degenerates to $\\Pr[A_q] < 1$, which every prime satisfies; it would certify a survivor at $H = x^2$ with the whole condition costing nothing (`import-suen.md` §8(i)). Two things are wrong with it, and both are on the record. First, the premise: exact independence of two residues on an interval of integer length $H$ needs $qq' \\mid H$, and for the whole family it needs $x\\# \\mid H$; $H \\ge x^2$ does not empty the graph (`import-shearer.md`, adversarial correction (3); carried into the proposal on 2026-09-06). Second, the quantifier: the lopsided hypothesis $\\Pr(A_q \\mid \\bigwedge_{q'\\in S}\\bar A_{q'}) \\le x_q\\prod_{q'\\sim q}(1-x_{q'})$ is required for every subset $S$ of the non-neighbours, and verifying it at $|S| = \\pi(x)$ needs equidistribution modulo $x\\#$, not $x^2$ (`import-suen.md` §8(i)). The exact criterion sharpens the diagnosis: on the pairwise-drawn graph with rule $qq' > H$, Shearer's threshold $H^*(x)$ is $35, 55, 65, 91, 115$ at $x = 13, 17, 19, 23, 29$ against $x^2 = 169, \\dots, 841$, an exponent $\\ln H^*/\\ln x$ near $1.41$ over seventeen levels with a proven floor $2/\\sqrt e = 1.2131$ (`import-shearer.md` §5; verified, floor proven there). So the pairwise-drawn argument does not merely reach the conjecture, it would prove a statement an order of magnitude stronger, and the defect is a hypothesis that is not being verified rather than a near miss.\n\n**(ii) The complete graph, and the Mertens wall as an identity.** If no conditioning set beyond a pair can be verified, the graph must be complete. The asymmetric local lemma's sufficient condition with weights $x_q = c/q$ needs $2 \\le c\\prod_{5\\le p\\le x,\\,p\\ne q}(1-c/p)$ for every $q$; the maximum of the right side over $c$ runs $5.00000, 1.25000, 0.85832, 0.71831, 0.63031, 0.57626, 0.53495, 0.50497$ at $x = 5, 7, 11, 13, 17, 19, 23, 29$ against the required $2$, feasible at $x = 5$ and at no larger level; optimising over all weight vectors rather than the ansatz does not change the verdict (`import-suen-01-transfer.js` PART D(ii); `import-shearer.md` §4; verified). The exact criterion is different. On a complete dependency graph the only independent sets are the empty set and the singletons, so the independent-set polynomial is $1 - \\sum_v p_v$ and Shearer's region is the simplex $\\sum_v p_v < 1$: the exact criterion is the union bound (Scott–Sokal, Example 3.1; proven, published). For the kill events that is $\\sum_{5\\le p\\le x} 2/p < 1$, which is $0.867532$ at $x = 11$ and $1.021379$ at $x = 13$. So the wall for any argument whose only inputs are the dependency graph and the marginals sits at $x = 13$ as an identity, not at $x = 7$; the $x = 7$ figure belongs to the sufficient condition, which on a clique is a factor $e$ short of the exact one (`import-shearer.md` §3, §4, §10 correction 2; the record's own first sentence on this point was corrected there). The witness needs no theorem: lay the $2K$ events as arcs of length $1/p$ end to end on the circle; the marginals and the within-prime exclusions are respected, and the arcs cover the circle exactly when $\\sum 2/p \\ge 1$, so a probability space consistent with both inputs has no survivor from $x = 13$ on (`import-shearer.md` §6, as corrected: the arcs are laid end to end, not as disjoint intervals with room to spare). This is the eighth independent arrival at the Mertens wall in the record's count (`sift-limit-attack.md` §7) and the first that is an identity.\n\n**(iii) An admissible repair is Bonferroni, which is Brun's pure sieve, and the two accountings on record are reconciled up to an endpoint convention.** Verify the hypothesis only for sets $S$ with $\\prod_{q\\in S} q \\le H$. A uniform cutoff $|S| \\le m$ with $x^m \\le H$ includes every such set, since $\\prod_{q\\in S}q \\le x^{|S|}$, but it is a sufficient device and not a size bound on each admissible set: at $x = 101$ and $H = 101^2$ the four primes $5, 7, 11, 13$ have product $5005 \\le 10201$ while $4 > 2$ (referee report 70 §5; the first draft's inference ran the wrong way). Inclusion–exclusion cut at that uniform depth $m$ is one conservative repair, not the only possible use of product-limited sets. The producer prices it with an even depth $m_0$, the smallest with $e_{m_0+1} < \\prod(1-2/p)$ for $e_j$ the elementary symmetric functions of $\\{2/p\\}$, and an error term taken as the largest single Stirling-bounded term, and prints $\\theta_{\\text{pure}} = 2.418, 3.864, 4.672, 5.915, 9.442$ at $x = 13, 101, 199, 1009, 10^6$, crossing $\\beta_2$ between $101$ and $199$ (PART D(iii)). The sift-limit record prints $\\beta_{\\text{pure}} = 2.80, 3.98, 4.25, 5.38, 8.88$ at the same levels, crossing between $227$ and $229$ (`sift-limit-attack.md` §7). The proposal flagged the two as unreconciled, with the sign of the discrepancy flipping across the range, and made reconciliation a downgrade trigger.\n\nThey are the same object under two error accountings, and the rigorous one is the sift-limit figure. This paper's script `job66-brun-pure.js` (§12) computes, for $n \\equiv 5 \\pmod 6$ in a window of $L$ consecutive values of $k = (n-5)/6$ and the primes $5 \\le p \\le x$, the pointwise Bonferroni lower bound at odd depth $m$,\n\n$$\\#\\{\\text{survivors}\\} \\ge L\\,A_m - B_m,\\qquad A_m = \\sum_{j\\le m}(-1)^j e_j,\\qquad B_m = \\sum_{j\\le m} 2^j\\binom{\\pi'}{j},$$\n\nwith $\\pi' = \\pi(x) - 2$; the error is one unit per residue class per squarefree modulus, and a modulus with $j$ prime factors carries $2^j$ classes by the Chinese remainder theorem. Any window with $L > B_m/A_m$ therefore holds a survivor, and $H_m = 6(B_m/A_m + 1)$ is a rigorous window length, $\\theta_m = \\ln H_m/\\ln x$. Minimising over odd $m$:\n\n| $x$ | 13 | 101 | 199 | 227 | 229 | 439 | 1009 | $10^6$ |\n|---|---|---|---|---|---|---|---|---|\n| best odd $m$ | 3 | 5 | 5 | 5 | 5 | 7 | 7 | 11 |\n| $\\theta$ (this paper) | 2.806 | 3.983 | 4.253 | 4.253 | 4.281 | 5.202 | 5.383 | 8.878 |\n| $\\beta_{\\text{pure}}$ (`sift-limit-attack.md` §7) | 2.80 | 3.98 | 4.25 | | | | 5.38 | 8.88 |\n| $\\theta_{\\text{pure}}$ (producer) | 2.418 | 3.864 | 4.672 | 4.642 | 4.661 | 4.836 | 5.915 | 9.442 |\n\n(Verified at the tabulated levels.) The same $A_m$ and $B_m$ are computed by the function `brunBound` of `research/attack-beta2-05-covering-prune.js`, with a larger depth cap, so the accounting identification is direct (referee report 70 §6 read that producer). The two exponents differ in an endpoint convention: the sift-limit record prints $\\ln(6B_m/A_m)/\\ln x$ and this paper $\\ln(6(B_m/A_m + 1))/\\ln x$, which at $x = 13$, $m = 3$ are $2.803954874890$ and $2.805711369726$ in exact rational arithmetic, so the two agree to two decimals and not to every printed digit, as the first draft said. Exact rational evaluation gives $4.252548612814$ at $x = 227$ and $4.280959847526$ at $229$, so the crossing of $\\beta_2 = 4.26645$ is between those primes, where the sift-limit record puts it; the scan finds no return below the threshold at any prime up to $2000$, which is a finite statement and not a proof for larger $x$ (the first draft's \"for good\" is withdrawn). The scan's depth cap of 25 and double-precision evaluation of the $e_j$ are disclosed; the exact rational checks cover $x = 13, 227, 229$ only. The producer's $\\theta_{\\text{pure}}$ is superseded as an accounting: its even-depth rule and largest-term error bound are neither an upper nor a lower bound on the rigorous quantity, which is why its discrepancy changed sign across the range. The identification of the local-lemma repair with Brun's pure sieve stands as an exponent benchmark, not as a numerical proof of an asymptotic sieve theorem. The exponent grows like $\\ln\\ln x$ (`sift-limit-attack.md` §7 states the constant, $7.182\\ln\\ln x$; this paper checks only the tabulated levels).\n\n**The address, stated once.** The blocking quantity is neither the dependency degree nor the expected number of kills. Degree is zero on the pairwise-drawn graph and the expected kill count $\\sum 2/q$ is $2.05$ at @23. What blocks is the admissible conditioning-set modulus: a budget $\\prod_{q\\in S} q \\le H$ is met by every set of size at most $\\ln H/\\ln x$, the hypothesis needs $|S| = \\pi(x)$, and every exponent in the table is that uniform cutoff priced (inferred). Exact CRT equidistribution is one sufficient route to the conditional bound the hypothesis asks for, not a necessary condition for every lopsided conditional inequality. The closure of the wider local-lemma family, Moser–Tardos, resampling oracles and Achlioptas–Iliopoulos, is `import-shearer.md` §8's and is not repeated here; its atomicity leg carries an unproven hypothesis and the closure survives on the sequel's unconditional $\\gamma_i \\ge \\mu(f_i)$ (adversarial note there). None of this excludes a use of arithmetic information beyond a graph and marginals, an alternative event representation, or a new conditional bound; what is closed is the graph-and-marginal interface, and on the resampling and variable-model interfaces the extra hypotheses are priced in that record.\n\n## 9. Prior art and what was searched\n\nThe corpus carried no mention of Janson, Suen, Harris or the local lemma before the import (`import-suen.md` §3). That sentence rests on a repository grep whose alternation was written in basic rather than extended syntax, so as recorded it searched for one literal string and could not have matched; re-running it correctly gives the same answer, but the recorded instrument does not establish it (`prop-suen-import.md` §4).\n\nWhat was searched. Five full texts on long prime gaps and sieved sets (Ford–Green–Konyagin–Maynard–Tao arXiv:1412.5029, Ford–Green–Konyagin–Tao 1408.4505, Maynard 1408.5110, Ford–Konyagin–Maynard–Pomerance–Tao 1802.07604, Banks–Ford–Tao 1908.08613) contain no occurrence of Janson, Suen, the local lemma or FKG; their probabilistic toolkit is Chebyshev, Hoeffding, Bennett and the Rödl nibble. The citation graphs of Suen 1990 and Janson 1998 were walked outward, 97 works on OpenAlex and 99 on Semantic Scholar, with none applying the family to coprimality or to a Jacobsthal-type problem, and the 2026 survey of the local lemma (Sason, arXiv:2603.07245) lists no number-theoretic application (`research/history/staging/proposals-prior-art.md` §4, officer pass of 2026-08-19; verdict there NOVEL-SO-FAR). Not searched: the citation graph of the lopsided lemma (Erdős–Spencer 1991; Lu–Székely), Google Scholar, MathSciNet review text, and the covering-systems literature as a body.\n\nFour near-misses cut the claim down. Hough's solution of the minimum-modulus problem for covering systems (Annals 2015) uses a relative local lemma, Hough–Nielsen (Duke 2019) use the cluster-expansion form, and Balister, Bollobás, Morris, Sahasrabudhe and Tiba (Invent. Math. 2022) then removed the lemma from the argument. Filaseta, Ford, Konyagin, Pomerance and Yu (JAMS 2007) write a lower bound for a sieving density as $\\prod(1-1/n_i)$ minus a sum over non-coprime pairs, a pair-correlation-corrected independence heuristic with Suen's shape and without the name; the record calls these authors FKMPT, which is the abbreviation the corpus uses for the 2021 JEMS paper of Ford, Konyagin, Maynard, Pomerance and Tao, a different set of authors (§10). Peres and Schlag (BLMS 2010) needed a bespoke one-sided local lemma for lacunary Diophantine approximation, and Peres and Yang (arXiv:2606.28860) cite Janson 1998 for a maximal-gap law of dilated lacunary sequences on the circle, the nearest methodological neighbour found. (Locators for Hough, Hough–Nielsen, BBMST, FFKPY and Peres–Schlag are as the record carries them, not re-verified for this draft; `research/PRIOR-ART.md` has no entries for these papers.)\n\nWhat is not claimed. `research/SEARCH-CONVENTIONS.md` §1 has a row for Shearer's region (owning convention: Shearer's region, the independent-set polynomial, the repulsive lattice gas) and no row for the probabilistic-combinatorics family itself (Janson's inequality, Suen's inequality, the local lemma, in the wording of Alon–Spencer and Janson–Łuczak–Ruciński). Under the corpus's rule that an absence must name the convention it searched, this paper asserts no absence and no novelty for the import. Writing that row and searching in it is the proposal's first upgrade trigger and is still open. Erdős and Spencer's 1991 paper was not read at source; its lopsidependency statement is taken from three concordant secondary sources (Alon–Spencer §5.6, Zhao's notes, Harvey–Vondrák arXiv:1504.02044 §1.3). Shearer's 1985 article text is closed and was not read; the identity §8(ii) needs is Scott–Sokal's Example 3.1, read at source in the record.\n\n## 10. Refuted and corrected claims\n\nRefutations stay visible. Items 1 to 5 are the record's own corrections of 2026-08-19, of which only item 4 has been applied; items 6 to 11 are corrections this paper's first draft made to the record; items 12 to 21 are corrections to that first draft made in this revision after referee report 70 and returns #973, #977 and #1043.\n\n1. `verify-cofactor-convolution.md` §3 (third bullet, \"a priori $O(\\sum_{q>x}q^{-2})$\") and §7 (\"$|\\delta| = 0.29\\%$ to $2.02\\%$ against a mutual-exclusion scale of $0.88\\%$ to $2.11\\%$\", read as a bound): refuted, $\\rho/F = 2.001$ at @11 and $1.266$ at @19 (§4). Not applied to that file.\n2. Same file, §9 bullet 4 (\"no proof that $\\delta = O(\\sum q^{-2})$\"): not an open lemma but a statement strictly stronger than the twin prime conjecture (§5). Not applied.\n3. `paper/anchored-note.md` §7 \"Conjectured sharp form\" and §5 \"the law's constants are HL-conditional\": the constant $e^{2\\gamma}/4$ is unconditional at the rung of §7 here; only $\\rho \\to 0$ is conjectural. Not applied.\n4. The closed-route row for bounding $\\delta$ by the forced scale: applied, `research/OUTCOMES.md` row of 2026-08-19, reworded 2026-08-29.\n5. The owning-convention row for the probabilistic-combinatorics family in `research/SEARCH-CONVENTIONS.md` §1: not written (§9).\n6. `import-suen.md` §5 and §1, \"Suen's inequality has exactly one level of content, and it is the smallest one\": with the printed neighbourhood convention Theorem 8's bracket is $-0.178$ with product inputs and $-0.491$ with anchored inputs at @11, so the content is none at every computed level and was none in the record's own product-measure setting (§6, verified).\n7. `import-suen.md` §3, \"$\\Delta = 0$ ... VERIFIED at seven levels; the OUTPUT block's $\\Delta_{\\text{match}}$ column\": the column is the constant $0$ written into `nuStats`; the statement is proven, not verified (§3).\n8. `import-suen.md` §5, \"no proper subgraph is admissible\", asserted from joint dependence of large sets: now verified at five levels by exact pairwise counts, with the correct reason (exact pairwise independence needs $qq' \\mid W$), and inferred beyond (§6).\n9. `import-suen.md` §8(iii) and the producer's readings 10: $\\theta_{\\text{pure}}$ is superseded by the rigorous Bonferroni exponent, which equals `sift-limit-attack.md` §7's $\\beta_{\\text{pure}}$; the crossing of $\\beta_2$ is between $227$ and $229$, not between $101$ and $199$ (§8).\n10. `import-suen.md` §11, near-miss 2: the 2007 JAMS authors are Filaseta, Ford, Konyagin, Pomerance and Yu, not the FKMPT of the 2021 JEMS paper (§9).\n11. `import-suen.md` §8(ii), \"it arrives earlier here ($x = 7$ rather than $x = 13$) because the LLL's condition is strictly stronger than the union bound\": true of the sufficient condition, false of the exact criterion, which is the union bound; already corrected in the record on 2026-08-19 and carried here (§8).\n\n12. First draft §6 and `job66-anchored-pairs.js`, and the producer's PART E: Theorem 8's weights excluded the endpoints of each edge, which Janson's definition of $k \\sim A$ does not allow; corrected values in §6 (referee report 70 §1; return #1043 quotes the printed sentence).\n13. First draft abstract, §6 and §10 item 6, \"the record's product-measure substitution had given it one level of content\": the positive value at @11 was the weight error; the corrected product-input bracket is $-0.178$ there and negative at all seven levels.\n14. First draft §6, \"the minimal admissible dependency graph is the complete graph minus at most two edges\", and the transfer to the grouped events: replaced by a lower bound on the edge set; the grouped inference is not drawn (§6).\n15. First draft abstract, §4 reading 1, §5 and §10 item 1, \"$\\delta$ is not bounded by the forced scale, refuted at two levels\": withdrawn; $|\\delta|/F < 1$ at all seven levels, and what is refuted is the sign (§4).\n16. First draft §4, remainder $O(\\rho F)$: corrected to $O(F^2 + |\\rho|F)$.\n17. First draft §5, \"$|\\delta| = O(F)$ is one-sided trivial\": withdrawn; the upper side from the dimension-2 sieve is $\\delta = O(1)$ (§5).\n18. First draft §1 and §5, \"the target cannot be reached by any inequality of the imported kind, because the target is the conjecture itself\": replaced by the implication actually proved (§5).\n19. First draft §8(iii), \"$\\prod_{q\\in S}q \\le H$ caps $|S|$ at $\\ln H/\\ln x$\": the inference is reversed; the cutoff is sufficient, not necessary (§8).\n20. First draft abstract and §8(iii), \"equals $\\beta_{\\text{pure}}$ to every printed digit\" and \"crosses for good\": the agreement is to two decimals under differing endpoint conventions, and the scan is finite (§8).\n21. First draft §2, \"the table is verified three ways at five levels\", and §11, the @31 resource estimate: the brute-force route exists at @11 and @13 only, and the estimate was wrong (event ids stored in a 16-bit array with $2K = 75{,}068$ at @31; the joint matrix alone about 22.5 GB; dense slot arrays on $2W/30$ entries exceeding 130 GB), so the present code cannot be launched at @29 or @31 (§2, §11).\n\nAlso on the record and carried: the producer's pre-registration PR2 (two to three orders of magnitude) is 1.4 orders at @11; the first draft of the producer omitted the self-strike clause and returned $S(11) = 47$; the import-shearer pre-registration's \"wall at $x = 7$\" half was refuted by two levels while its \"boundary is the Mertens threshold\" half was upgraded to an identity.\n\n## 11. What would move it\n\n- **The pair counts beyond @23.** The first draft's estimate for @29 and @31 was wrong (§10 item 21): the check script's 16-bit event ids overflow at @31, its joint matrix there is about 22.5 GB, and its dense slot arrays exceed 130 GB. A streaming or compressed implementation is needed before §6's pair table can be carried to @29 and @31; the qualitative Theorem 8 verdict at those levels already follows from the product-input brackets, which are negative there, and from the regime condition.\n- **$\\delta$ at @37 and @41.** $S(37)$ and $S(41)$ are on record; $L_0$ and $R_0$ there are prime counts on the comb above $y$ below $7.4\\times10^{12}$ and $3.0\\times10^{14}$, within reach of a segmented sieve on donated compute. They decide the pre-registered prediction of §4.\n- **A referee for Lemma 4.** The assembly is written; a check against the corpus's sieve conventions, or by a reader who knows the fundamental lemma's error terms, would move the constant from inferred to proven and regrade `prop-anchored-note.md` with it.\n- **The upper-bound constant of §5**, run through the comb normalisation, so that \"$\\delta \\le C'$\" carries a number.\n- **The owning-convention row of §9**, written and searched; it is the proposal's cheapest trigger and the only one that concerns the absence claim.\n- **Erdős–Spencer 1991 at source.**\n\nDowngrade triggers as pre-registered: a positive $\\delta$ at @37 or @41; an exhibited conditioning set whose failure behaves differently from the account in §8, which would retire the wall section. The reconciliation trigger of the proposal is discharged by §8(iii) up to the stated endpoint convention.\n\n## 12. Reproduction\n\nThe record's producer and this paper's three scripts. Served paths are relative to the project's documents base; this paper's files are uploaded with its return and identified by sha256 there.\n\n| script | what it produces | invocation | time |\n|---|---|---|---|\n| `research/import-suen-01-transfer.js` | §2 counts at seven levels, §3 $F$ and $\\Delta$ columns, §4 table, §6 product-measure Suen table, §8 PART D tables | `node research/import-suen-01-transfer.js 11,13,17,19,23,29,31` | 576.1 s (embedded; out-sha256 `eb59337f…`) |\n| `job66-anchored-pairs.js` (first draft) | §2 counts at five levels by an independent method; §6 pair table; its Theorem 8 columns use the endpoint-excluded weights and are superseded | `node job66-anchored-pairs.js 11,13,17,19,23` | 6 s, one core, about 0.5 GB |\n| `t8check1427.py` (this revision) | §6 Theorem 8 quantities with the printed neighbourhood convention, one event per prime and the $2K$ orientation events, anchored and product inputs, with the $\\varphi_3$, Remark 6 and Suen variants; reproduces referee report 70's and return #973's values | `python t8check1427.py 11 13 17 19` (numpy) | 2 min, 1 GB at @19 |\n| `job66-sieve-prediction.py` (this paper) | §7 finite-level table; brute-force $L_0$, $R_0$ at @11 and @13 | `python3 job66-sieve-prediction.py` (numpy) | 1 s, 0.5 GB |\n| `job66-brun-pure.js` (this paper) | §8(iii) table and the crossing scan to $x = 2000$ | `node job66-brun-pure.js` | under 1 s |\n\nThe producer was re-run for this revision at its default levels @11 to @23 (run-suen.out, uploaded) and matches the embedded block line for line there. The Node scripts and the Python scripts are deterministic and their logs are hashed in the return; the logs carry timing lines that differ run to run and should be excluded from a byte comparison. The Shearer figures of §8(i) and (ii) are `research/import-shearer-01-region.js` (15.9 s, embedded).\n\n## References\n\nLocators are those the research record carries unless marked; a locator identifies the evidence and does not mean the cited page was re-read for this draft unless stated.\n\n- N. Alon and J. H. Spencer, *The Probabilistic Method*, Wiley, 2nd ed. 2000: Theorem 8.1.1 (Janson's inequality), Theorem 8.7.1 (Suen's inequality), Theorem 6.3.2 (Harris/FKG), §5.6 (the lopsided local lemma). Read as page images in the record (2026-08-19).\n- S. Janson, *New versions of Suen's correlation inequality*, Random Structures & Algorithms 13 (1998) 467–483: Theorems 1, 2, 3, 8, 9; Remarks 2, 3, 4, 6; the definition of $k \\sim A$ on printed page 2 (\"$i \\in A$ is not excluded\") and the regime sentence for Theorem 8 on printed page 5. Read from the author's copy at Uppsala (www2.math.uu.se/~svantejs/papers/sj121.pdf, SHA-256 5f6a04aa5578e93cb81e1df1f91c8e4e44aed2b8ea5c945bca436eaaf1596c76, text layer and page images agreeing; returns #1043 and referee report 70).\n- K. Ford, *Sieve Methods Lecture Notes*, Spring 2023, ford126.web.illinois.edu/sieve2023.pdf (SHA-256 a6e8462f1e76606614e5c2891b419515be408d5f11f0b82915f5c24e05c00e06): Theorem 3.4 (Bombieri–Vinogradov, printed page 35), Theorem 3.6 (fundamental lemma, printed page 38). Read from the text layer for this revision.\n- Platform records revised against: referee report 70 on return #22; returns #973 (corrected brackets at five levels), #977 (restricted-graph triage), #1043 (the printed convention and the regime condition).\n- W. C. S. Suen, *A correlation inequality and a Poisson limit theorem for nonoverlapping balanced subgraphs of a random graph*, Random Structures & Algorithms 1 (1990) 231–242 (locator from Janson's reference list).\n- S. Janson, T. Łuczak and A. Ruciński, *Random Graphs*, Wiley, 2000, Theorem 2.23.\n- P. Erdős and J. Spencer, *Lopsided Lovász local lemma and Latin transversals*, Discrete Appl. Math. 30 (1991) 151–154. Not read at source (paywalled, no open copy); statement from Alon–Spencer §5.6, Y. Zhao's MIT lecture notes Theorem 6.5.1, and Harvey–Vondrák §1.3.\n- O. Riordan and L. Warnke, arXiv:1203.1024 (Janson's inequality for arbitrary up-sets in a product space).\n- N. J. A. Harvey and J. Vondrák, *An algorithmic proof of the Lovász local lemma via resampling oracles*, arXiv:1504.02044, §1.3.\n- L. Lu and L. A. Székely, Electron. J. Combin. 14 (2007) #R63.\n- J. B. Shearer, *On a problem of Spencer*, Combinatorica 5 (1985) 241–245 (abstract read in the record; text closed).\n- A. D. Scott and A. D. Sokal, *The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma*, J. Stat. Phys. 118 (2005) 1151–1261 = arXiv:cond-mat/0309352v2, Example 3.1, Theorem 4.1 (read as the arXiv PDF in the record).\n- E. Bombieri, *On the large sieve*, Mathematika 12 (1965) 201–225; A. I. Vinogradov, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 903–934 (as recorded in `research/bv-import-survey.md` §1).\n- H. Halberstam and H.-E. Richert, *Sieve Methods*, Academic Press, 1974 (the classical reference for the fundamental lemma and the sieve form of Bombieri–Vinogradov; not opened; the statements used are taken from Ford's notes above).\n- H. Diamond and H. Halberstam, *A Higher-Dimensional Sieve Method*, Cambridge Tracts 177 (2008), Table 17.1: $\\beta_2 = 4.26645\\dots$; the twenty-place value is Booker and Browning's ancillary data (`research/SEARCH-CONVENTIONS.md` §4).\n- R. Hough, Ann. of Math. 181 (2015); R. Hough and P. Nielsen, Duke Math. J. 168 (2019); P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, Invent. Math. (2022); M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu, J. Amer. Math. Soc. 20 (2007), Lemma 2.1; Y. Peres and W. Schlag, Bull. London Math. Soc. 42 (2010); Y. Peres and B. Yang, arXiv:2606.28860 (2026); I. Sason, arXiv:2603.07245 (2026). Locators as the record carries them; not re-verified for this draft.\n\n## Authorship and AI disclosure\n\nSole author: Chris Benjaminsen. The framework, vocabulary, and driving questions are the author's, developed over six years of independent work. Formal derivations, literature audits, computations, and manuscript drafting were carried out using AI assistants under the author's direction. Computations have reproducible code and recorded outputs; asymptotic arguments require their stated mathematical inputs and are not proved by finite checks. Refuted intermediate claims are retained in the record.\n\n"}