# Smooth sieve literature: existing results and the next signed question **The sufficient twin-prime margin remains OPEN.** This is a bounded source audit and one application of the matched upper-bound machinery, not an originality claim. The source-informed priority is now to estimate the signed contribution of specified factor configurations with a paid tail. Further generic moment calculations or profile optimization must name what they buy for that signed estimate. Use the exact C3 profile, floors and notation of [global-smooth-majorant.md](global-smooth-majorant.md): T=log x, J_x=(x/2,x], L_i=log(b_i/a_i), (a_L,b_L)=(floor(x^.22),floor(x^.24)), (a_R,b_R)=(floor(x^.04),floor(x^.05)), W_i=b_i. The complete identity is \[ S(x)=C_2x+\widehat{\mathcal R}(x)+O_A(x/\log^A x),\qquad \widehat{\mathcal R}(x)=\sum_{n\in J_x} \widehat G_L(n)\widehat G_R(n-2). \tag{1} \] ## 1. What the sources already do | Source and inspected location | Reusable content | Limit for this campaign | |---|---|---| | Granville--Koukoulopoulos--Maynard (GKM), [v4](https://arxiv.org/html/1606.06781v4), section 1.2, Theorems 1.3--1.4 and 10.4, Lemma 10.3 | Finite differences; moment and small-prime concentration estimates; integral transfer between profiles | One-point even moments do not determine the signed shifted pair. Smoothness requirements depend on the moment. | | Carneiro--Chirre--Helfgott--Mejia-Cordero (CCHM), [v6](https://arxiv.org/html/2005.03162v6), Theorem 1.2, Corollary 1.3 | Two-parameter quadratic main term and profile optimality | Optimizes the one-point quadratic form, not the full absolute pair or its sign. | | Green--Tao, [Appendix D](https://arxiv.org/html/math/0606088), Lemma D.2, Theorem D.3 and proof | Smooth shifted divisor correlations; explicit sieve factors | Its divisor theorem allows infinite-complexity systems; its prime theorem does not thereby apply to twins. Smoothness and support must still match. | | Goldston--Yildirim III, [section 8](https://arxiv.org/html/math/0209102), (8.1)--(8.10) | Different truncation levels, constants and product-of-levels error | The displayed theorem concerns truncated Lambda sums, not prime-filtered Ghat. Uniformity must be proved before integrating a family of cutoffs. | | Henriot, [original](https://arxiv.org/html/1102.1643v1) and [2014 erratum](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B0BD208D979495FE06B8192209E710EA/S0305004114000280a.pdf/nair-tenenbaum-uniform-with-respect-to-the-discriminant-erratum.pdf), corrected New Theorem 5 | Upper bound for a nonnegative function in a uniform growth class, without multiplicativity | Not a signed theorem or a sharp constant. New Theorem 6 has additional hypotheses. | The source convention and lookup scope are recorded in [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md). Searches used smooth sieve weights, smallest prime factors, shifted divisor correlations and Nair--Tenenbaum bounds. This is not a complete citation-tree review. ## 2. Profile optimization already has an answer for the quadratic norm Set h(t)=chi(1-t), D_1=a_i and D_2=b_i in CCHM. The source hypotheses hold: h is absolutely continuous, constant outside [0,1], and h' has bounded variation. Expanding the finite divisor sum and counting multiples gives an O(b_i^2) error since |rhohat_i|<=1. Consequently, \[ \sum_{n\in J_x}\widehat F_i(n)^2 =\frac{|J_x|}{L_i}\int_0^1|\chi'(t)|^2\,dt +O(x/L_i^2+b_i^2) =\frac{700|J_x|}{429L_i}+O(x/L_i^2+b_i^2). \tag{2} \] Here chi'(t)=-140t^3(1-t)^3, so the integral is 19600 B(7,7)=700/429. The [validator](smooth-sieve-literature-validation.js) checks it separately by polynomial integration and the factorial beta formula. For every absolutely continuous transition with the same endpoint values, integral |chi'|^2 >= |integral chi'|^2=1. The linear taper attains one. Thus our C3 choice increases this norm's leading constant. Its purpose was to enable a different shifted absolute estimate. Smoother does not mean better under every objective, and the quadratic optimization problem should not be reassigned without changed constraints or a changed objective. Equation (2) concerns Fhat, not Ghat. It does not identify the implicit constant of the Henriot majorant or compare that constant with C2. ## 3. GKM suggests localization, but its one-point bound is not enough GKM Theorem 10.4(a) bounds the moment weighted by the number of small prime factors; Theorem 1.4(a) gives the corresponding exceptional-set bound. With the exact profile map in global-smooth-majorant section 7, the C3 hypothesis admits second and fourth moments: 3>1 and 3>6/4. It does not meet the general-profile sixth-moment condition 3>20/6. The proof via Lemma 10.3 exposes bounded derivative norms, which stay bounded for our family because log(b_i)/L_i stays bounded. For 0=2, the second-moment source bound reads \[ \sum_{\substack{n\le x\\P^-(n)\le x^\delta}} \widehat F_i(n)^2\ll \delta x/T. \tag{3} \] Applying Cauchy to (3) and the other side's full second moment only gives O(sqrt(delta) x/T) for the absolute Fhat pair. Multiplication by the two rough-part logarithms then gives O(sqrt(delta) x T). That calculation loses a logarithm relative to the required scale. It is not the shifted localization result below. ### 3.1. A symmetry condition needed when reusing the prime integrals GKM Lemma 10.5 is stated for general C1 functions, but replaces an ordered prime sum by 1/m! times a full-cube integral. The proof's first equality needs permutation symmetry. This omission is present in v4, pp. 68--69, and in the [journal version, pp. 1158--1159](https://smf.emath.fr/system/files/filepdf/ens_ann-sc_54_1089-1177.pdf). Here is an asymptotic counterexample to the unrestricted statement. Take m=2 and the fixed bounded C1 function g(t_1,t_2)=tanh(t_2-t_1), with y=exp(Y), z=exp(2Y). The cube integral is zero by antisymmetry. Every ordered summand is positive. Restrict p to (exp(Y),exp(5Y/4)] and q to (exp(7Y/4),exp(2Y)]; then g(log p,log q)>=tanh(Y/2). Mertens' prime harmonic asymptotic therefore gives a positive liminf at least log(5/4)log(8/7) as Y tends to infinity. The displayed lemma's error tends to zero, since g and its gradient are bounded and both harmonic interval factors remain bounded. Thus the unrestricted statement is false. The paper's ensuing integrand g_m^(2k), built from all subset sums, is symmetric. This defect does not invalidate that application or the moment statements used above. For a weight that privileges the smallest prime, retain the ordering, or establish an ordered-domain integration formula with its own error; do not import the cube/m! formula unchanged. Our harmonic proof in section 5 uses ordered sums and inequalities and does not use Lemma 10.5. ## 4. What shifted divisor correlations do and do not reopen Green--Tao Theorem D.3 requires that the affine forms are not rational multiples. The forms n and n-2 qualify; only p=2 is exceptional. The theorem explicitly does not require finite complexity. Lemma D.2 gives c_(f,1)=-f'(0) and c_(f,2)=integral_0^infinity |f'|^2. Thus a smooth plateau at zero has zero first-order sieve factor. This explains why cancellation in an unweighted short-divisor surrogate can be classical even at a fixed shift. A direct import for our exact C3 profile is not asserted. The theorem uses smooth compactly supported functions and a sufficiently small cutoff exponent. A compact extension on the negative axis is harmless for positive divisors, but it does not turn C3 into C-infinity. The proof uses rapid Fourier decay; replacing it by finite regularity requires its own error check. Nor may a fixed exponent .24 be called sufficiently small without examining the support calculation. There is a useful changed support fact. Direct CRT expansion of pure F_L^r F_R^s has a counting error O(b_L^r b_R^s), for fixed nonnegative integers r,s and bounded divisor coefficients. Each compatible system has one residue class modulo the lcm and hence an O(1) interval error. For r=s=1 the exponent is .29; for r=s=2 it is .58. Both are below one. The earlier transition calculation had a product above x. Its failed raw-support import does not exclude these smaller objects. This counting observation alone does not evaluate the main form. For the earlier linear taper there is the exact identity \[ F_i^{\rm lin}(n)=\frac{\Lambda_{b_i}(n)-\Lambda_{a_i}(n)}{L_i}, \qquad \Lambda_R(n)=\sum_{d\mid n}\mu(d)\log(R/d)_+. \tag{4} \] Goldston--Yildirim's unequal-level theorem is therefore the existing tool for this surrogate. Its singleton constant C_1=1 makes the four leading terms cancel for two distinct shifts and fixed exponents. For our smoother profile, two integrations by parts give instead \[ \widehat F_i(n)=\frac1{L_i}\int_0^1 \chi''(u)\Lambda_{a_i e^{L_i u}}(n)\,du,\qquad \int_0^1\chi''(u)\,du=0. \tag{5} \] Indeed chi(v)=integral_0^1 chi''(u)(u-v)_+ du for v>=0; for v<0 the right side is one because integral chi''=0 and integral u chi''=1. This also covers d=1 and both constant portions of the profile. Uniform source errors on the compact exponent rectangles, interval endpoints and floors must be handled before consuming (5). Even a completed pure-F calculation omits H_L H_R and the prime exclusions. Neither paper supplies those extra weights by allowing arbitrary bounded multipliers. The actual signed Ghat consumer remains open. ## 5. A shifted small-prime bound from the matched upper theorem This is a derived application, motivated by section 3. It changes the majorant, not the profile or identity (1). Fix 00, partial summation of the Mertens prime sum gives \[ \sum_{p\le z}\frac{(\log p)^b}{p-1}\ll_b(\log z)^b. \tag{8} \] This pays the omega(m)=1 and 2 terms in sum_(m<=x) v(m)/m; their normalization powers are T^(1-sigma) and T^(2-sigma). For omega(m)>=3, fix its first three primes p0. This retains all additional factors and their powers, rather than truncating them. Together with m=1 it proves sum v(m)/m=O_sigma(1). **Shifted import.** Reuse Henriot's corrected New Theorem 5 with X=Y=x/2, Q_1(t)=t, Q_2(t)=t-2, alpha=1/2, delta_source=1, epsilon_source=1/1200. The polynomial-size and degree hypotheses, fixed resultant and corrected density bound <=2/(ab) are unchanged. The latter follows from the compatible congruences a|n, b|n-2, whose density is gcd(a,b)/(ab), with gcd(a,b)|2. Equations (7)--(9) and the harmonic bound for w therefore give \[ \sum_{n\in J_x}\bigl(v(n)w(n-2)+w(n)v(n-2)\bigr) \ll_\sigma x/T^2. \tag{10} \] If P^-(m)<=x^delta, then w(m)=ell_1^sigma v(m)<=delta^sigma v(m). Applying this on either side of the pair, with a union bound, yields \[ \sum_{\substack{n\in J_x\\P^-(n(n-2))\le x^\delta}} |\widehat F_L(n)\widehat F_R(n-2)| \ll_\sigma\delta^\sigma x/T^2. \tag{11} \] The constants are independent of delta. If x^delta<2, the set is empty. The prime two, both possible marked sides, m=1 in the source class, and squareful integers have all been accounted for. The absolute exceptional-set proof of [global-factor-signs.md](global-factor-signs.md) section 2 also bounds the error on any subset. Since each H_i=log t_i is at most T and dropping composite filters enlarges a nonnegative sum, it gives \[ \boxed{\displaystyle \sum_{\substack{n\in J_x\\P^-(n(n-2))\le x^\delta}} |\widehat G_L(n)\widehat G_R(n-2)| \ll_\sigma\delta^\sigma x +O_\epsilon(x^{39/40+\epsilon}).} \tag{12} \] Fix epsilon<1/40. No signed improvement or explicit small constant is inferred from (12). It is a localization of the existing absolute budget. ## 6. The remaining factor problem and the prime-filter trap Let R_delta be the Ghat pair restricted to P^-(n(n-2))>x^delta. For sigma fixed, (12) implies \[ |\widehat{\mathcal R}-R_\delta| \le K_\sigma\delta^\sigma x +O_\epsilon(x^{39/40+\epsilon}). \tag{13} \] At fixed delta, each remaining integer has Omega(n)<=1/delta, counting multiplicity. Thus the full remainder can be approximated to any fixed multiple of x by finitely many prime-factor types. This is not a density theorem for those types: their prime variables still satisfy the coupled equation n-(n-2)=2. For delta<.04, primes in these configurations can still lie on both sides of each a_i,b_i boundary. The ten-prime negative Fhat cell has not disappeared. Its sign must be retained whenever its factors lie above the new cutoff. Removing prime exclusions is particularly dangerous here. Put A_i(m)=Fhat_i(m)H_i(m) and P(m)=1_(m prime)log m. On the present dyadic interval A_i(p)=log p for primes, so C_i^comp=A_i-P exactly. Consequently \[ \begin{aligned} C_L^{\rm comp}(n)C_R^{\rm comp}(n-2) ={}&A_L(n)A_R(n-2)-P(n)A_R(n-2)\\ &-P(n-2)A_L(n)+P(n)P(n-2). \tag{14} \end{aligned} \] The last term is the weighted twin count itself. Accounting for proper prime powers converts it to S with the existing negligible correction. Hence evaluating unfiltered surrogates and then treating all filter terms as already known would assume the problem being investigated. This is a specific algebraic warning, not a theorem excluding every use of a filtered or signed sieve. There is also an essential quantifier limit. A fixed delta gives a fixed error K_sigma delta^sigma x. It cannot automatically absorb cx/log^K x as x grows. A proof retaining only this error would need a sufficient fixed linear surplus, or a uniform analysis with delta=delta(x) shrinking, or a sharper signed estimate of the tail. When delta shrinks the number of allowed prime factors grows. We have neither a numerical K_sigma nor evidence that exhaustive enumeration at a useful delta would be computationally practical. ## 7. Revised next step and conditions for reuse The [executed switching match](switching-negative-mass.md) prices a three-prime/rough-composite family using ordinary BV and the linear sieve, including an upper bound for the excluded prime partners. It derives liminf Nhat/(C2*x)>3/2. A negative-only upper bound below C2*x is therefore refuted for the current cutoff exponents, including the localized remainder at fixed delta<1/50. The next comparison must retain positive mass; the net signed margin is still OPEN. 1. Keep (2), the source moment thresholds and the pure-divisor correlation formulas as imported background. Do not spend another attempt deriving their orders without a new signed consumer. 2. Use (13) to write the actual opposed-sign factor contribution, retaining the positive part and prime exclusions. Match each proposed estimate to classical Buchstab decomposition, switching or almost-prime sieve results before inventing a replacement. Read [chen-opportunity-audit.md](chen-opportunity-audit.md) first: finite factor count and a clipped weight were already insufficient there. 3. The next bounded output should be a matched comparison of positive and negative families, with its constant and complement budget. State which joint prime-factor density remains an open input. If the computed or derived bounds leave a deficit, record that scoped failure instead of enlarging an unvalidated computation. For example, a fixed-delta estimate R_delta >= (-C2+K_sigma delta^sigma+c)x for some c>0 would suffice via (1), (13). It is stronger than necessary and is presently OPEN. Merely restating it is not progress. PNT estimates for individual prime variables do not supply the coupled factor density. ## 8. Source custody and validation Read on 2026-09-06: the locations in section 1, GKM's section 10.3 proof and Green--Tao's Fourier/Euler-product argument in Appendix D. This is not a claim to have independently re-proved all imported theorems or read every section of every paper. The theorem pages and their continuations were inspected as rendered PDFs as well as text. The existing Henriot correction and source-class match were reused and checked against the new weights in section 5. | PDF | SHA-256 | Principal page locators checked | |---|---|---| | GKM v4 | 67feecce65ced4250aa046d6e0c36300744eb84358d21d53409a512d1649e34d | PDF pp. 9, 66--68; source equations (1.6)--(1.7), Theorems 1.4, 10.4 | | GKM published version | 287f0f9953370ff5dc874217babdd6679619951bc005ab4c95fb84425a94765d | Printed pp. 1158--1160, PDF pp. 72--74; symmetry check for Lemma 10.5 and its application | | CCHM v6 | 0bd3729efe1c89edf14c1ca22240af8362d6375397505cbcbb557dfe296eb309 | PDF pp. 4--5, Theorem 1.2 and Corollary 1.3 | | Green--Tao published PDF | ec126f9c9e189cc0308b83fb4f9f93c29a47d11ad875391aff4ce4c42aef1001 | Printed pp. 1828--1835, PDF pp. 78--85 | | Goldston--Yildirim III | 6be1422119bf78f43ea265da70bc73ce6b952ca07560f80b5105d96a81eca66f | PDF pp. 30--31, section 8 | | Henriot erratum | bd56f8f411457549c5cc53f9b24bbdec68544dca09cfe0836e35cc1b25677d6f | Printed pp. 375, 377; custody in global-smooth-majorant section 4 | [smooth-sieve-literature-validation.js](smooth-sieve-literature-validation.js) checks the rational derivative constant, squared v-envelope growth at sigma=1/2, small-prime suppression with exact ordered proxies, squareful inputs and (14). Controls reject the same growth argument with a negative first-coordinate exponent, a C3 sixth-moment import and the omission of the twin term. An antisymmetric control detects the ordered/unordered substitution in Lemma 10.5, while a symmetric control verifies its permitted use. The first filter check used JavaScript numbers and tripped over signed zero; using BigInt makes that integer identity exact without a signed-zero convention. A failure of uniform growth, (9), the corrected theorem hypotheses or the absolute exceptional-set transfer would invalidate (12). Those are written arguments; the finite validator does not establish an asymptotic, an effective onset, a useful constant or a twin margin.