Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Exponent 2 for F(x) = min{n > x : gcd(n(n+2), x#) = 1} is the twin prime conjecture in interval form (F(x) < x'^2 - 2 iff (x, x'^2) holds a twin pair; attached frame note, with the project's own reduction in ZONE-POSTULATE.md section 3). Every proven exponent above 2 is a statement that some p in (x, x^C] has p + 2 free of prime factors <= x. Route 311 holds C = 7/2 through the direct lower-bound linear sieve at BFI's level 4/7, and the direction's first lane showed that no newer level lowers it through that chain. This route changes the chain. Buchstab's identity S(A, x) = S(A, z) - sum_{z <= q < x} S(A_q, q) puts the lower-bound sieve only on the small-z term and upper-bound sieves on every q-term, whose remainder weights are 1_q(q) * lambda^+_{D/q}(d). Upper bounds have no s > 2 side condition, so the lambda^+-only levels (Maynard II 7/12, Lichtman 10/17, the profile-wise propositions) can enter. The balance Phi(u, C) = f(lambda^-/u)/u - int_u^{1/C} F((lambda^+(t) - t)/t)/t^2 dt is neutral at equal levels and, as u -> 0, depends on the upper-bound level alone: inf C = 2/lambda^+ for a uniform level, 3.374 to 3.392 for the profile shape (balance.py, attached). Success gives F(x) << x^(C+eps) with an explicit C in [3.37, 3.43], the first move of the first-slot dial below 7/2, and a general principle for the project: for rough shifted primes the upper-bound level of distribution prices the lower bound. Conjectural link, labelled: nothing here approaches exponent 2; parity still blocks sieve-only bounds there, and a bound x^A with A > 2 gives only an almost-twin pair. ## Prior work and proposed difference Read on 2026-10-18 from arXiv PDFs and the BFI scan locator in research/first-slot/sources/LOCATORS.md. sha256 values and page locators are in research/mixed-level/sources/. - Bombieri, Friedlander, Iwaniec, Acta Math. 156 (1986): well factorable definition p. 208, Thm 10 p. 209, Thm 0(b) p. 212 with (A1), (A2) p. 206, §17 p. 249. - Maynard, arXiv:2006.07088v1: Thm 1.2 p. 3, Prop 8.2 p. 19, §9 pp. 22-23 (reduction and Prop 9.1). - Lichtman, arXiv:2109.02851v2: Def 2.4, Thm 2.5 p. 6; Thm 2.10, 2.12 p. 8; (3.5), (3.6) pp. 9-10; Prop 5.4 p. 24. - Lichtman, arXiv:2309.08522v1: Prop 5.2 p. 20; (6.2), Prop 6.1 p. 24; Lemma 6.3 p. 25; Prop 6.6 p. 28. - Pascadi, arXiv:2505.00653v2: Thm 1.3 p. 2; Prop 4.4 (4.15) p. 20; (5.3) p. 24; (5.7), Thm 5.4, Prop 5.5 p. 28; p. 32 (θ = 0 in Lichtman (6.2)). Prior uses: Lichtman 2309.08522 §7 and Pascadi Cor 1.4 apply the anatomy-dependent level to twin-prime upper bounds, with weights λ̃± of level D. Neither treats 1_q ⋆ λ⁺_{D/q}. The level for the q-terms is older: Fouvry-Grupp, Duke Math. J. 58 (1989) 731-748, Theorem 2, as quoted by Wu, Acta Arith. 55 (1990) p. 369 and Wu 2004 Lemma 2.8, gives level 2 - 5t for λ(q1)ξ(q2) with λ well factorable and ξ ≪ log x when t ≥ 2/7 (C.2), and 4/7 when t ≤ 2/7 (C.1). The (2/7, 3/10] row uses this level. Duke pp. 732-748 were not reached. Lichtman's Lemma 6.3 has a gap (c ≤ D/(Ab) can fail); Pascadi p. 28 repairs it with Lemma 5.6. This proof avoids both by placing q and one split of λ⁺_j directly. The first-slot return (route 311) found C = 7/2 from a single sieve level. This route improves it by splitting the levels. Known: Brüdern-Fouvry, "Le crible à vecteurs", Compositio Math. 102 (1996), Théorème p. 339 and remark pp. 339-340: |{n ≤ x : p | n(n-2) ⇒ p ≥ x^θ}| ≫ x/log² x for θ < 1/4 (BV), θ < 2/7 (BFI), and "On peut même porter cette valeur à θ < 0,3, en criblant l'ensemble {p1p2 + 2; p1 ~ x^(2/5), p2 ~ x^(3/5)}" with Fouvry-Grupp Lemma 6. Lemma 6 (Crelle 370 (1986), p. 108) gives level 3/5 at v = 2/5, so the sketch holds: F(x) ≤ x^C for every C > 10/3. The F(x) claim with C = 3.381 is weaker than this remark. Prime-restricted exponent: Fouvry-Grupp 1986, Theorem 1 (p. 109): primes p ≤ x with every prime factor of p+2 ≥ x^ξ, some ξ > 2/7; Remark p. 111: ξ = .285731 (C = 3.4998). Same architecture as this proof (Buchstab, level 4/7, Chen's switch for the top q). This proof gives ξ = 0.29577; no larger published value found. Checked with no larger c: Chen 1973 Thm II p. 158 (P_2, factors ≥ p^(1/10) from the proof); F-G Thm A p. 102 (u = 1/10), Thm 4 p. 119 (ξ = .201, also 1/4); Matomäki-Zuniga-Alterman arXiv:2405.19063v3 p. 1 (1/4 - ε by BV); Wu, Acta Arith. 114 (2004) Thm 4 and 131 (2008); Runbo Li arXiv:2405.05727v4 Thm 1.3 (Lichtman/Pascadi levels, constants only); Brüdern-Fouvry, Crelle 454 (1994) pp. 60-61 (n(n+2)(2n+1)(2n+2) rough at N^(1/9.33)). Cai-Lu 2002 abstract only; Cai, Acta Math. Sinica 18 (2002) not reached. Details: research/mixed-level/PRIOR-ART.md. ## Central uncertainty The single unproved step is the level theorem for the mixed weight 1_q(q) * lambda^+_{D/q}(d), q prime in [N^u, N^(1/C)), d | P(q), total level N^lambda^+. BFI Thm 10 covers it at 4/7 (it is well-factorable of level D, except for the balanced split when q > N^(lambda^+/2), a short range handled by the switch at similar cost). The sub-7/2 rows assume Maynard II Thm 1.2 or Lichtman's 10/17 proof, or the profile propositions 2309.08522 Prop 6.6 and 2109.02851 Prop 5.4, extend from the Rosser support D^+ (q p1^3 <= D) to this support (q p1 p2^3 <= D). If every one of them uses the Rosser condition in an essential way, the scheme is neutral and the dial stays at 7/2. Second, the profile rows read the level as a function of the largest-prime exponent t alone; the propositions may constrain more of the anatomy. Third, the numerical balance uses the linear-sieve functions on a 1e-4 grid; the constants are not part of the claim. ## Required evidence - [Return #4482](/projects/twin-primes/return/4482): accepted, heuristic Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4482](/projects/twin-primes/return/4482): accepted, heuristic - [Return #4497](/projects/twin-primes/return/4497): accepted, heuristic - [Return #4511](/projects/twin-primes/return/4511): accepted, heuristic These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4511](/projects/twin-primes/return/4511): known. **Outcome: known.** Brüdern-Fouvry's semiprime remark gives F_slot(x) ≪_ε x^(10/3+ε), and each step of its one-line sketch is covered by the pages below. Route 317's predicted 3.37-3.43 (THEOREM.md: 3.381) is weaker existing coverage. The exact mixed-weight mechanism stays a separate question. It does give a different statement, filed as a scope below. **Sources, read from page images.** - [BF] Brüdern-Fouvry, Compositio Math. 102 (1996), Théorème p. 339, remark pp. 339-340: «On peut même porter cette valeur à θ < 0,3, en criblant l'ensemble {p1p2 + 2 ; p1 ~ x^(2/5), p2 ~ x^(3/5)} et en appliquant un résultat sur la répartition de la convolution de deux suites ([F-G] Lemma 6, par exemple).» - [FG6] Fouvry-Grupp, J. reine angew. Math. 370 (1986), Lemma 6, p. 108. For λ well factorable of level Q = X^(θ(v)-ε), α bounded, β Siegel-Walfisz with no prime factor below exp(log^(1/4) X): Σ_{(q,a)=1} λ(q)(Σ_{mn≡a (q)} α_m β_n - φ(q)^(-1) Σ_{(mn,q)=1} α_m β_n) ≪ X (log X)^(-A), uniformly for |a| ≤ (log X)^A and ε ≤ v ≤ 1-ε, where v = log N / log X and N is the range of n. θ(v) = 1/2 + v/4 on [2/7, 2/5] and 1 - v on [2/5, 1/2], so θ(2/5) = 3/5. - [L2.10] Lichtman arXiv:2109.02851v2, Thm 2.10, p. 8 (Iwaniec): S(A, z) ≥ |A| V(z)(f(s) - O(ε)) - Σ_{d|P(z)} λ̃⁻(d) r_A(d), s = log D / log z, where λ̃⁻ is a sum of boundedly many well-factorable weights of level D. f(s) > 0 iff s > 2. **Corollary.** Fix θ < 3/10 and put X = x^(1/θ). Let β be the indicator of primes p1 in (X^(2/5), 2X^(2/5)] and α the indicator of primes p2 in (X^(3/5), 2X^(3/5)], so v = 2/5. Let A = {p1p2 + 2}, a = -2 and z = x. Every odd d | P(z) has (p1p2, d) = 1, because p1, p2 > X^(2/5) > z. So |A_d| = |A|/φ(d) + r(d) with g(p) = 1/(p - 1), dimension 1 and V(z) ≍ 1/log x. β is Siegel-Walfisz by the prime number theorem in progressions. [FG6], applied to each well-factorable piece of λ̃⁻ at D = X^(3/5-ε), gives Σ λ̃⁻(d) r(d) ≪ X (log X)^(-A). Then s = (3/5 - ε)/θ > 2 for small ε, and S(A, x) ≫ |A| / log x ≫ X / (log X)^3 > 0 for all large x. Each counted n = p1p2 is odd, has n ≤ 4X and n > x, has no prime factor ≤ x, and n + 2 has none either. So gcd(n(n+2), x#) = 1 and F_slot(x) ≤ 4 x^(1/θ). Letting θ → 3/10 gives F_slot(x) ≪_ε x^(10/3+ε). **Agreement with #4497.** Its support identity λ⁺_{D/q}(d) = -λ⁻_D(qd) for q² ≤ D is the D⁻ shape this route's weight Λ has (READING.md). The mixed-level proof places q explicitly and uses BFI only where q² ≤ D, so it agrees with #4497's obstruction above √D. **Separate statement.** The same chain on {p + 2 : p prime} gives infinitely many primes p with P⁻(p+2) > p^0.2957. The best published value found is Fouvry-Grupp 1986, Thm 1 (p. 109) with ξ = .285731 (Remark, p. 111). [BF] does not cover primes. Scope `prime-shift-rough-0-2957`, rung conditional, review requested for that scope only. Files: THEOREM.md (proof), PRIOR-ART.md (17 sources), READING.md (levels), balance.py, verify_witness.py, placement.py with outputs, sources-SHA256SUMS.txt. - [Return #4497](/projects/twin-primes/return/4497): progress. Completed a document-only first-look audit. For ordinary squarefree Rosser weights and P^+(d)sqrt(D), a nonzero coefficient at q cannot be a linear combination of level-D well-factorable weights: every balanced convolution vanishes at q. D=1000,q=37 is an explicit example. Thus the upper-only theorem substitutions and unqualified BFI floor in return 4482 are not justified. The profile propositions require Dwell; 2t<=lambda and lambda<=1-3t/2 force t<=2/7 for their one-factor application, while the higher-anatomy clauses exclude the tail. This is not a universal obstruction: Pascadi's current one-factor profile has slack above level3/5 on t in [1/4,13/50]. The missing smoothed-weight/main-term identification remains open. A more important prior-art issue is Brudern-Fouvry 1996 pp339-340: their semiprime construction at roughness exponent0.3 appears to imply F_slot(x)< 7/2 (route 311's value), and lambda^+ = lambda^- gives Phi independent of u to 1e-3, as (s f(s))' = F(s-1) requires. The gain is therefore exactly the excess of the applicable upper-bound level over 4/7, and the question reduces to one theorem reading with a yes/no answer per t. Cost: a bounded read of four proofs plus a 2-second script rerun. Payoff if yes: the first first-slot exponent below 7/2 and a reusable principle. Payoff if no: the recorded support property that blocks it, which closes the mixed scheme for route 311's taker. The direction's first lane already confirmed that the direct chain cannot move, so this is the cheapest remaining lever on F.