# Separating the small-common-divisor kernel, and what the Kloosterman-fraction interfaces cost **No new region is controlled by this note, and twin-prime infinitude remains OPEN.** The result is a derived separation of the remaining kernel plus three priced negative interface tests. The global consumer [grouped-divisor-moment (20)](grouped-divisor-moment.md) is unchanged: E_dagger, its cut set (19) and W_dagger are exactly as before. A failed upper bound is a failure of the stated estimate, not evidence about the size or sign of the underlying correlation. Throughout, the box is (delta,nu)=(8/25,9/20) from [RESEARCH-HANDOFF §5](RESEARCH-HANDOFF.md), so the expanded lengths are M<0, uniformly over boxes, bands, endpoints, gcd branches and the divisor twists. The moment route bounds |T| by (sum_m |A_left(gm)|^2)^(1/2) times the square root of the moment in [grouped-divisor-moment (1)](grouped-divisor-moment.md), the first factor being O(M^(1/2) log x). The moment's four classes have these **block** exponents at this box: | class | source | block exponent | margin below 1 | |---|---|---|---| | equal frequency R=0 | (5) | 39/50 | 11/50 | | complete periods, all j | (11) | 14/25 | 11/25 | | nonzero R, j>x^(1/20) | (12) | 397/400 | 3/400 | | nonzero R, j<=x^(1/20) | (10) | 103/100 | deficit 3/100 | Only the last exceeds one, and its majorant on a dyadic band j~J is N^3/J^(3/2). Summing that geometrically, the single band j in [1,2) already carries 1/(sum_{k>=0}2^(-3k/2))=64.6% of the majorant, and no restriction j<=J_1 with J_1 fixed lowers the exponent below 3/2. **The residual pressure is a coprime-pair problem, not a small-common-divisor problem**: the whole 3/50 moment deficit is present at j=1 alone. The label "small common divisor" describes where (12) stops, not where the mass sits. ## 2. The exact completed kernel, with the coupled weight displayed For an ordered pair set j=(u_1,u_2), u_i=j*ell_i, (ell_1,ell_2)=1, c=j*ell_1*ell_2, R=h_1*ell_2-h_2*ell_1, r=-theta*R, and F(m)=Phi_{u_1,h_1}(m)*conj(Phi_{u_2,h_2}(m)). Expanding both endpoint differences gives the exact four-term form \[ F(m)=\sum_{\epsilon_1,\epsilon_2\in\{0,1\}} (-1)^{\epsilon_1+\epsilon_2} e\!\left(\frac{\lambda_{\epsilon_1\epsilon_2}}{m}\right),\quad \lambda_{\epsilon_1\epsilon_2} =\frac1g\left(\frac{h_1z^{(\epsilon_1)}}{u_1} -\frac{h_2z^{(\epsilon_2)}}{u_2}\right), \tag{2} \] with z^(0)=z_0', z^(1)=z' the two endpoints, |z^(i)|<=x. Each |lambda|<=0, \[ |\widehat W(it)|\ll_j\frac{\min(1,|\kappa|Y)(1+|\kappa|Y)^j}{(1+|t|)^j}, \qquad \int_{-\infty}^{\infty}|\widehat W(it)|\,dt\ll f. \tag{5} \] Truncating at |t|<=(1+v)x^epsilon leaves O(x^(-100)), and on Re(s)=0 \[ y^{-s}=h^{-it}m^{it}u^{it}. \tag{6} \] So the endpoint weight separates into **unimodular** factors attached to h, m and u. Three consequences, all used below. 1. Separation in u is free up to x^epsilon, even though u sits in the denominator of the phase: (6) separates the product mu multiplicatively, and u^(it) leaves |b_u| and ||b||_2 unchanged. 2. The only cost is the truncation height (1+v)x^epsilon and the L1 mass f. At this box v<=1 on every retained band, so the cost is x^epsilon and the gain f is retained on the lower bands. This is the same (1+v) and f bookkeeping as [grouped-divisor-moment (6)](grouped-divisor-moment.md); it is cheap here only because A<=MN/x. 3. The independent divisor twists d^(-s), (de^3)^(-t) of [residual-coverage §4](residual-coverage.md) and [grouped-divisor-moment §5](grouped-divisor-moment.md) have nonnegative real part, so they multiply b_u and A_left(gm) by factors of modulus at most one. Any interface that sees the coefficients only through their L2 norms is therefore automatically uniform in the twists and in the twist heights. After (5)--(6) the block (1) is, up to x^epsilon and the factor f, the pure trilinear Kloosterman fraction \[ \mathcal T=\sum_{m\sim M}\sum_{u\sim N}\sum_{h\in H} \alpha_m\beta_u\nu_h1_{(m,u)=1}\, e\!\left(\frac{\sigma\theta h\bar m}{u}\right), \tag{7} \] with ||alpha||_2<1.} \tag{11} \] The second bracket term gives only 399/400<1. **The first term is the whole failure**, by 4/125. For comparison the classical budget is 103/100 and the trivial bound 53/50, so at this box Bettin-Chandee is weaker than classical completion by 1/500 and short of what is needed by 4/125. DFI (1.1) alone, with the harmonic band summed trivially, gives 1267/1200, weaker still. **What would have to change.** Holding the other exponent fixed, replacing (M+N)^(1/4) by (M+N)^kappa closes this box iff kappa<27/140, and replacing (AMN)^(7/20) by (AMN)^gamma closes it iff gamma<9/28. Note 9/28<1/3: even an (AMN)^(1/3) first term would not suffice. **Region.** The BC conditions at the top band are (7/10)(a+b)+(1/4)max(a,b)<17/20 and (7/8)(a+b)+(1/8)max(a,b)<1. Over 19005 rational boxes covering delta in [6/25,19/25], nu in [1/20,19/20], the 3218 boxes BC controls are **all** inside the 7004 already controlled by delta+nu<19/25 or 5delta+2nu<123/50 or (delta<19/25 and delta+3nu<161/100); 3786 already controlled boxes lie outside BC, and none lie outside the existing region. So this interface **adds no region at all**. **Two later papers, both unusable here.** [Dong-Robles-Zeindler, arXiv:2601.00292](https://arxiv.org/abs/2601.00292) claimed a 1/12 balanced saving in place of DFI's 1/48; the paper stands at v2 (2026-01-05) with an author erratum recording that "a factor of L^2 was accidentally missed in equation (2.53)", so the claimed improvement does not follow; the authors state the rest of the argument is still valid. The improvement must not be imported. [Wright, *Trilinear Kloosterman fractions I*, arXiv:2604.25177v2](https://arxiv.org/abs/2604.25177), Theorem 2.1, improves BC when the denominator carries a **fixed** factor R, bounding the same sum over (m,nR)=1 by an asymmetric bracket with a leading R^(1/4). Read only through a fetched rendering, so treat as a secondary reading: at R=1 its bracket prices to block exponent 1023/1000 at this box, still above one. Our modulus u is an arbitrary expanded divisor with no forced factor beyond g in {1,2}; a canonical smooth-part factorisation u=Ru' does produce a fixed R, but the R=1 class survives and its bound is the R=1 bound, so nothing is gained. ### 5B. Kuznetsov and the spectral large sieve The audit's diagnostic ([structural-literature-audit §3A](structural-literature-audit.md), Pascadi arXiv:2404.04239v3, Theorem 3; translation inherited from that note, not re-derived here) puts L=N/J, H=A, K asymptotic to HL at level q asymptotic to L^2, with the extra factor comparable to max(1,H^2/L) when H<=L. Here H^2=x^(3/25) and L=x^(1/2)/J, so the factor is trivial for every J<=x^(19/50). That contains the entire remaining range J<=x^(1/20) with a wide margin: **the structured spectral advantage is absent exactly where the deficit sits**, and present only where (12) already suffices. A Kuznetsov application independent of that theorem needs complete Kloosterman sums whose coefficients do not depend on the modulus. Both available completions fail that at this box, and the failure is explicit rather than inherited: * completing u modulo m in (7) gives (1/m)*sum_{t mod m} hatbeta_m(t) S(theta h,t;m) with hatbeta_m(t)=sum_u beta_u e(tu/m): the coefficient depends on the modulus m; * completing m modulo c in (3) gives hatF(t) and S(t,r;c) where c is itself the summation variable, and both hatF and r=-theta R depend on the factorisation c=j*ell_1*ell_2. Naming a specific DI theorem number here would be an assumed rather than a checked dependency: [Deshouillers-Iwaniec, Invent. Math. 70 (1982)] was not reachable in this session. The obligation is stated structurally and must be discharged against the primary text before any such import. ### 5C. Composite-modulus bilinear Kloosterman sums [Blomer-Pascadi, arXiv:2607.24311v1](https://arxiv.org/abs/2607.24311), Theorem 1.1, and [Pascadi, GAFA published version](https://link.springer.com/article/10.1007/s00039-026-00746-0), Theorems 1.1-1.2, require a bilinear form in complete Kloosterman sums with two separated coefficient sequences on intervals of length B<=c. The audit's instruction was to reach §3B only for a separated short bilinear subexpression. **No such subexpression appears.** Equation (7) carries no complete Kloosterman sum; the only one available is in (3), where the pairing is (t mod c) against r, the t-interval has full length c, and the modulus c=j*ell_1*ell_2 is itself one of the summation variables. At B=c the displayed savings in Theorem 1.1 are larger than one. The balanced factorisation c=ell_1*ell_2 with both factors near x^(1/2) is genuinely present at j=1 and is the one structural feature of this box that Theorem 1.2 is built for; the missing input is a short separated pair of coefficient sequences against a fixed modulus, not the factorisation. Constructing one, if possible, is the only route in §3B that this note does not close. ## 6. Region of validity and effect on the global consumer The derived items are (8)-(10), the correction-phase bound in §4(b), the separation (5)-(6), and the interface pricings (11) and §5B-§5C. The separation results hold on any expanded box with A<=MN/x, that is v<=1, for both gcd branches, both endpoint conventions, all coefficient sectors and uniformly in the divisor twists; the constants depend only on epsilon and on the fixed power bounding the lengths. The pricings are exact rational statements at (delta,nu)=(8/25,9/20) and, for the region claim, on the stated rational grid. **Effect on E_dagger: none.** The cut set [grouped-divisor-moment (19)](grouped-divisor-moment.md) and the reduction (20) are unchanged, and (21) remains OPEN. The controlled region is still delta+nu<19/25 or 5delta+2nu<123/50 or (delta<19/25 and delta+3nu<161/100), with the uniform product threshold still every fixed exponent below 19/25. Controlling this rectangle would not by itself prove the sufficient global margin of [RESEARCH-HANDOFF §3](RESEARCH-HANDOFF.md), and failing to control it proves nothing about the sign or size of the correlation. What is reusable is the reduction itself: the block is now a pure trilinear Kloosterman fraction (7) with unimodular twists and known norms, so any future DFI-type bound plugs straight in, and the exact thresholds it must beat are kappa<27/140 or gamma<9/28. ## 7. Validation and falsifiers [small-divisor-kernel-validation.js](small-divisor-kernel-validation.js) checks the reciprocity law as an integer identity, the index-by-index split (9) modulo m and as exact rationals, the complex form of (8), the endpoint interchange (10) in both directions, the size and m-variation of the correction phase on a model box, and every rational exponent in §5 including the region containment. Negative controls reverse the sign in the reciprocity law, use the product u_1u_2 in place of the lcm, drop theta, use the wrong sign in the correction split, and replace the existing region by one of its three conditions; the last produces 722 containment violations, so the containment test is not vacuous. The deterministic artifact is [small-divisor-kernel-validation.json](small-divisor-kernel-validation.json). These checks verify identities and arithmetic, not the imported papers' proofs and not any asymptotic rate. What would change the conclusions: a corrected reading of Bettin-Chandee Theorem 1 with different exponents; a first-term improvement past 27/140 or 9/28; a modulus factorisation exposing a genuinely short separated bilinear form for §5C; a Kuznetsov formulation whose coefficients are modulus-independent; or an error in the separation, in which case (7) is not the right reduction and §5 prices the wrong object.