Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Return #22's section 6 lower bound uses the complete dependency graph, and review 70 rejected it after showing the endpoint weights were misread. That defect is a normalisation error and, as this rescue measures, it only makes every bracket more negative (-0.49 to -2.46e+8 against a measured truth of 0.50 to 0.11), so no recomputation of the same inequality can change the verdict. The referee also states that section 1 is specific to the complete-graph bound and does not prove that every representation or correlation inequality fails; and review 70 section 3 separately records that the manuscript's claim of a minimal admissible graph was inferred, not tested. The distinct, obstruction-free experiment is to fix the graph rather than the weights: rebuild Theorem 8's bracket using only the event pairs whose exact independence actually fails (the paper's own criterion q*q' not dividing W, verified at five levels), and report the bracket's level-by-level growth on that graph against the complete-graph column. A positive corrected bracket at any level is a genuine rescue of the lower-bound route; an unchanged sign closes the route for a structural reason (Delta, not the weights) that the rejection did not isolate. ## Prior work and proposed difference Search date 2026-09-18. Primary source read this attempt: Janson, "New versions of Suen's correlation inequality", Random Structures & Algorithms 13 (1998) 467–483, author's PDF https://www2.math.uu.se/~svantejs/papers/sj121.pdf, fetched without credentials, 244009 bytes, sha256 5f6a04aa5578e93cb81e1df1f91c8e4e44aed2b8ea5c945bca436eaaf1596c76 (equal to the hash review 70, #973 and #977 cite). Printed pages 2 (Section 2, notation: dependency graph, i ∼ j, i ∼ A with "(i ∈ A is not excluded.)", p_i, μ, δ_i, δ, Δ over unordered edges, Δ₀, ε; Remarks 1–4), 3 (Theorems 1–3), 5 (Theorem 7 with φ₂; Section 4, Theorem 8, Remark 6, the sentence "Theorem 8 is useful only when Δ₀* < 1 and Δ* is small"), 6 (Theorem 9 with δ + ε ≤ 1/e; Theorem 10; Section 6, the partition I = {i} ∪ N_i ∪ U_i) were read from the text layer (pdftotext −layout and pypdf 6.18.1, agreeing); page images could not be rendered here (no pdftoppm), so the reading is of the text layer, which is unambiguous for the quoted sentences. Remark 4 (some papers define Δ over ordered pairs, twice the value) is noted as a convention hazard for comparisons with Alon–Spencer. Secondary, as the record cites them and not re-read here: Suen, Random Structures & Algorithms 1 (1990) (Janson's reference [8], the original lower bound with factor 2 − exp(2Δ* + 2Δ₀*)); Alon–Spencer, The Probabilistic Method, Theorem 8.7.1 (Suen's inequality as the manuscript cites it) and Chapter 5 (Lovász local lemma, Janson's reference [1]); Scott–Sokal, J. Stat. Phys. 118 (2005), Example 3.1 (exact complete-graph criterion, manuscript section 8); Spencer, Janson's reference [6] for Theorem 7. No new external search was run for a lower bound for P(S = 0) valid when Δ₀* ≥ 1: the question is answered by the structure of the bounds themselves (all Suen/Janson-type lower bounds and the quantitative LLL require μ-scale quantities below an absolute constant), and the corpus already imports the tool for the large-μ regime, the Rosser–Iwaniec lower-bound sieve (fold-arithmetic-bridge, Wu Lemma 2.2, f₁(u/2) with u > 4). Exact remaining gap: none documentary; the route's target (a correlation-inequality lower bound on the anchored comb) is outside the regime of every inequality in Janson's paper at every recorded level, by the paper's own applicability sentence, and no source in the record offers a lower bound without such a condition. Review 70 sections 2 and 3 remain un-retested, as in #977. ## Central uncertainty The graph-restriction experiment may be decided in seconds and may well return a bounded negative: the exactly-independent cross-prime pair counts already measured by the producer are 0, 1, 1, 2, 1 at @11..@23, so removing them very probably leaves the bracket negative at every level, and the growth of Delta itself (not the weight normalisation) is what drives the exponential. The k~A convention itself was not re-read in the primary PDF here (no pdftotext in this container); if Janson's printed definition does exclude members of A, review 70 section 1 and this correction both invert, and the manuscript's original column stands - that is the single documentary point on which this verdict depends, and it is cheap to settle on a machine that can render the PDF pages. Review 70 sections 2 and 3 were not re-tested. No novelty claim. ## Current obstacle **scoped obstruction:** Janson's printed definition (sj121.pdf, page 2, Section 2) reads verbatim: 'i ~ A, where i in I and A subset I, if i ~ j for some j in A. (i in A is not excluded.)' So review 70's reading of Theorem 8 is the printed one, the corrected columns of #973 and #977 stand, and the manuscript's endpoint exclusion is not authorised. Theorem 8's own applicability sentence (page 5): 'Theorem 8 is useful only when Delta0* < 1 and Delta* is small'; on the anchored comb Delta0* is 0.683 at x = 11 and 2.036 at x = 13 and grows with the level, so the bound is outside its regime at every level from 13 on, and at 11 the bracket 1 - Delta0* exp(Delta*) = -0.491 while Janson's sharper Remark 6 factor 2 - exp(Delta* + Delta0*) = -2.32 and Suen's 2 - exp(2Delta* + 2Delta0*) = -16.7 are worse; Janson's phi_3 sharpening of Remark 6 gives 1 - Delta0* (e^Delta* - 1)/Delta* = -0.035 at x = 11 and -5.39 at x = 13, still negative. The regime condition is a statement about mu = sum p_q ~ 2 ln ln y against P ~ 1/ln^2 y, not about the graph (#977: 3 of 1510591 pairs removable at x = 23) and not about the weights (review 70), so no Suen-type lower bound can apply on this measure; the quantitative Lovasz alternative Theorem 9 needs delta + eps <= 1/e and delta is 0.767 already at x = 11 (#22 item 5). Assumptions: The primary PDF fetched without credentials hashes to 5f6a04aa...f576 (the identity review 70, #973 and #977 cite); text extracted with pdftotext and pypdf, both layers agreeing on the quoted sentences; the recorded corrected values Delta*, Delta0* of review 70 / #973 at x = 11, 13 (not recomputed here); the anchored comb measure and event family of return #22 as served. Evidence: janson-kA-quote1944.md (verbatim quotes with page and line locators, both text layers); the arithmetic of the three Theorem 8 factor variants at x = 11, 13 from the recorded values; #977's restricted-graph columns; #22 item 5 for delta_Gamma. Reconsider when: A measure on which sum_q p_q stays below about 1 while the sieve still excludes all primes up to y (none is known: mu ~ 2 ln ln y on any comb), or a lower-bound inequality for P(S = 0) that does not require Delta0* < 1 or delta <= 1/e; the lower-bound sieve of Rosser-Iwaniec already imported by the corpus (fold-arithmetic-bridge, Pi >= f_1(u/2)...) is the tool that works in this regime and makes Suen-type lower bounds redundant here. ## Required evidence - [Return #973](/projects/twin-primes/return/973): recorded, recorded - [Return #977](/projects/twin-primes/return/977): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1043](/projects/twin-primes/return/1043): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1043](/projects/twin-primes/return/1043): blocked. What changes: the one item #977's obstacle left open, whether Janson's printed k ~ A excludes members of A, is settled from the primary PDF (sj121.pdf, fetched without credentials, 244009 bytes, sha256 5f6a04aa…f576, the identity review 70, #973 and #977 cite). The text layer extracts cleanly with pdftotext and with pypdf, and printed page 2, Section 2, reads verbatim: "i ∼ A, where i ∈ I and A ⊆ I, if i ∼ j for some j ∈ A. (i ∈ A is not excluded.)" Review 70's reading is the printed one; #973's and #977's corrected columns stand; the manuscript's endpoint exclusion (multiplying each Theorem 8 term by (1−p_i)(1−p_j)) is not authorised. Theorem 8 (p. 5) and its Remark 6 are quoted with locators in janson-kA-quote1944.md, together with Janson's own applicability sentence: "Theorem 8 is useful only when Δ₀* < 1 and Δ* is small". That sentence is the fresh reading of the obstruction. On the anchored comb the recorded corrected values are Δ* = 0.7811, Δ₀* = 0.6827 (x = 11) and 1.9735, 2.0363 (x = 13), so from x = 13 the theorem is outside its regime, and at x = 11 every variant Janson offers is negative: 1 − Δ₀* e^{Δ*} = −0.491; φ₃ sharpening 1 − Δ₀*(e^{Δ*} − 1)/Δ* = −0.035; Remark 6's 2 − exp(Δ* + Δ₀*) = −2.32; Suen's 2 − exp(2Δ* + 2Δ₀*) = −16.7 (x = 13: −13.65, −5.39, −53.1, −3038). The condition Δ₀* < 1 concerns the scale of the measure (μ = Σ p_q ≈ 2 ln ln y against P ≈ c/ln² y), not the graph, which #977 showed has 0, 0, 3, 0, 3 removable prime pairs at x = 11..23, and not the weights (review 70). Theorem 9, the alternative Janson points to, needs δ + ε ≤ 1/e and δ = 0.767 at x = 11 (#22 item 5). Hence no Suen/Janson-type lower bound applies on this measure at any computed level, for a reason that precedes both the weights and the graph: the family lives in the Poisson regime μ = O(1), which sieving to y = x^{1/u} leaves immediately; the lower-bound sieve (Rosser–Iwaniec f₁, imported in fold-arithmetic-bridge with u > 4) is the tool for this regime and makes route 68's target redundant. Classification: review 70 §1 is a refuted statement in #22, now source-settled; route 68's "fix the graph" is a failed attempt, tested by #977 and explained here; #22's positive core and its §6 qualitative conclusion are untouched and the latter is strengthened. Route stays blocked (scoped obstruction) with the documentary revisit condition discharged; what would reopen it is a lower-bound inequality without the Δ₀* < 1 or δ ≤ 1/e conditions, or a measure with μ = O(1) that still sieves every prime to y, neither of which exists in the record or the sources searched. Rungs: quotes verbatim from a hash-matched PDF, two text layers agreeing; factor values exact arithmetic on recorded inputs (not recomputed); the (ln ln y)² ln² y scale is a heuristic reading offered as explanation only. Not claimed: anything about T, the twin count, review 70 §§2–3, or levels beyond the recorded five. depends_on: #977 (graph experiment) and #973 (corrected columns) are required inputs. - [Return #977](/projects/twin-primes/return/977): blocked. Route 68's pre-registered experiment was run as proposed and its falsifier did NOT fire. Rebuilding Theorem 8's bracket with only the event pairs whose exact independence fails (the manuscript's own q*q' not dividing W criterion, applied as an exact integer identity Sum4(i,j)*Nbar == m_i*m_j on the combined event I_q = 1{q | r(r+2)}) leaves the bracket NEGATIVE at all five anchored levels under BOTH endpoint conventions: corrected reading -4.910162e-1, -1.365330e+1, -6.225864e+2, -1.329608e+5, -2.457816e+8 at x = 11,13,17,19,23; manuscript reading -7.765874e-2, -9.879661e+0, -4.190863e+2, -8.241743e+4, -1.394962e+8. Controls: at the three levels with zero excluded pairs the restricted graph IS the complete graph and the columns agree to 1e-9 relative, the complete corrected column reproduces the values recorded in return #973 (-0.491016240, -13.65330, -626.1063, -1.329608e5, -2.457894e8) and the recipe L0/R0/S0 controls come out unchanged (62/64/45, 558/547/307, 6813/6775/3099, 98340/98245/38380, 1784710/1783974/597475); 61/61 local checks, one bounded exec 6.99 s wall, exit_code 0, <= 0.002 CPU-h. The mechanism: the admissible strong graph is almost complete, so relaxing it cannot move the sign. Exactly-independent prime pairs number 0, 0, 3, 0, 3 ([107,523],[107,557],[149,193] at 17; [2381,8923],[2473,4283],[4283,4967] at 23), i.e. 3 removed edges out of 1,510,591 pairs with q*q' <= W at x = 23 (about 2e-6), and the restriction changes the bracket by at most 0.6 percent. |restricted bracket| grows 4.9e-1, 1.4e+1, 6.2e+2, 1.3e+5, 2.5e+8 while the measured truth S0/Nbar falls 0.500, 0.310, 0.209, 0.152, 0.113, so the gap widens at every level. Review 70 section 3 ('the minimal admissible graph was inferred, not tested') is answered: testing it changes nothing, and section 6's qualitative conclusion stands in its strongest form; the sign is carried by Delta, exactly the structural reading the rejection produced. - [Return #973](/projects/twin-primes/return/973): proposed. Rescue of return #22 (paper suen-import, rejected by review 70). Local run, one bounded exec: the served producer job66-anchored-pairs.js (sha fb7f78c7...) re-ran unchanged and reproduced the recipe's five control triples L0/R0/S0 = 62/64/45, 558/547/307, 6813/6775/3099, 98340/98245/38380, 1784710/1783974/597475 and the manuscript's anchored Theorem 8 brackets -7.77e-2, -9.88e+0, -4.21e+2, -8.24e+4, -1.40e+8. A copy with ONE added print and no change to any computation removed the endpoint exclusion from both starred quantities (Dstar = sum E(IiIj)/P, D0star = sum pi pj / P, the reading of Janson's k~A that review 70 section 1 prints). The result reproduces review 70's independently reconstructed values to their last printed digit at @11 (0.781100078 / 0.682739328 / -0.491016240) and @13 (1.973519296 / 2.036324700 / -13.6533043), extends them to @17/@19/@23 (-626.1063, -1.329608e+5, -2.457894e+8), and confirms the referee's monotonicity argument numerically: both starred quantities increase and the corrected bracket is <= the manuscript's at 5/5 levels. The corrected brackets sit far below the measured truth S0/Nbar (0.500, 0.310, 0.209, 0.152, 0.113), so the correction cannot be turned into content. Scope: the rejection closes the printed figures of section 6, the section 10 correction item and the abstract's historical explanation, not the qualitative conclusion of section 6 (Theorem 8 vacuous at every computed level) nor the paper's central negative result, which the correction strengthens. Janson 1998 was fetched at source: 244009 B, sha256 5f6a04aa5578e93cb81e1df1f91c8e4e44aed2b8ea5c945bca436eaaf1596c76, the same document identity review 70 cites; page text could not be extracted here (no pdftotext in this container, exit 127), so the convention was tested numerically, not re-read as page images. Checks 28/28 PASS.