# The squarefree coefficient core and the sparse exceptions **The sufficient twin estimate remains OPEN.** This note simplifies a further rectangle of the remainder; [prime-dispersion.md](prime-dispersion.md) now establishes cancellation of that entire rectangle by consuming this reduction. The controlled rectangles in [endpoint-fourier.md](endpoint-fourier.md) and [endpoint-pairing.md](endpoint-pairing.md) remain available. The latter also improves the generic QQ Fourier budget by keeping endpoints paired. There are two outcomes. The nonsquarefree coefficients are sparse enough to remove more endpoint terms. The squarefree coefficients do not have a uniform power saving in their individual norms. Further progress on the remaining term must use more than that proposed norm improvement. All asymptotic statements below are written derivations using the stated inputs, separate from the finite validation. ## 1. Exact formulas on a dyadic divisor interval For I=(D,2D] and integer W>=1 use A_0,A_1 from endpoint-fourier.md (4). The contribution at an expanded divisor l is A_0(l)log n+A_1(l). Define $$ L_0(\ell)=\mu(\ell)1_I(\ell),\qquad L_1(\ell)=-\mu(\ell)1_I(\ell)\log\ell, $$ $$ Q(\ell)=\mu(\ell)1_{\ell>2D} \sum_{p\mid\ell,\ p\le W,\ \ell/p\in I}\log p, \qquad B(\ell)=A_1(\ell)1_{\mu(\ell)=0}. \tag{1} $$ The p sum is over primes. Then, exactly, $$ A_0(\ell)\log n+A_1(\ell) =L_0(\ell)\log n+L_1(\ell)+Q(\ell)+B(\ell). \tag{2} $$ For squarefree l only r=p can survive in the convolution defining A_1, and mu(l/p)=-mu(l). If l is in I, then l/p<=D for every p>=2, so its prime sum is empty. Outside I the nonzero squarefree terms lie above 2D and have the form Q. All terms inside Q(l) have the same sign. If l has two distinct repeated prime factors, removing a power of one prime cannot make l/r squarefree, so B(l)=0. Otherwise write l=p^a m with a>=2, m squarefree and p not dividing m. Only r=p^(a-1) and r=p^a can survive. Their signs give $$ B(p^a m)=\mu(m)\log p\left( 1_{p^{a-1}\le W}1_I(pm)-1_{p^a\le W}1_I(m)\right). \tag{3} $$ The two interval indicators cannot both be one: if m>D, then pm>2D. Thus each nonzero B is exactly plus or minus log p. Its sign is not constant, and it is not identically zero. For I=(5,10], W=4, the exact witnesses are B(12)=-log 2 and B(28)=+log 2. The low terms have support at most 2D. Q and B have support in (2D,2DW]. These supports and squarefreeness separate the cases, and the original bound |A_1(l)|<=2log l can be sharpened to |A_1(l)|<=log l. This constant improvement itself does not change a power budget. ## 2. A smaller norm for the nonsquarefree part Count representations l=d p^k that can contribute to B, allowing overcounting. For k=1, d must be divisible by p. There are at most $$ \sum_{p\le\min(W,2D)} \left(\lfloor2D/p\rfloor-\lfloor D/p\rfloor\right) \ll D\log(2W) \tag{4} $$ such representations. Indeed the summand is at most D/p+1<=3D/p for p<=2D; the harmonic sum over all integers suffices. For k>=2 there are O(sqrt(W)log(2W)) possible prime powers and at most D choices of d for each. Consequently $$ \#\operatorname{supp}B\ll D\sqrt W\log(2W),\qquad \|B\|_2\ll D^{1/2}W^{1/4}\log^{3/2}(2DW). \tag{5} $$ These are global support and norm bounds, so they hold after restriction to any expanded-divisor box or to l=gm with g in {1,2}. They also hold for |B| in the Fourier error majorant. On a box of length M use the smaller of (5) and the pointwise bound O(sqrt(M)log x). No cancellation between different B coefficients is assumed. ## 3. Why the squarefree norm cannot gain a fixed power Let D=floor(x^delta), W=floor(x^w), for fixed delta,w>0. Define the top band T=(DW/2,2DW]. The following lower bound is derived, not fitted: $$ \sum_{\ell\in T}|Q(\ell)|^2 \gg D W\log W. \tag{6} $$ To prove it, take p in (W/2,W], d in I, d squarefree and p not dividing d. Then l=dp belongs to T, is squarefree, and contributes log p to the positive magnitude inside Q(l). Repeated representations of l cannot cancel: (sum log p)^2>=sum (log p)^2. Thus $$ \sum_{\ell\in T}|Q(\ell)|^2 \ge \sum_{W/22D. Therefore the count in (7) is at least a fixed positive multiple of D, uniformly for these primes once x is large enough. Ordinary PNT gives sum_{W/20 is possible. T is the union of two multiplicative dyadic boxes, so a uniform such norm saving on all boxes is also excluded. This closes a **specific norm-only improvement**, not cancellation in the two-coefficient phase sum. It does not rule out logarithmic gains, special ranges, coprimality effects or a one-sided estimate. ## 4. Insert the sparse norm into the complete endpoint budget Keep x dyadic, V=floor(x^(6/25)), Z=floor(x^(1/20)). Use I=(D,2D], J=(E,2E], with D=floor(x^delta), E=floor(x^nu), delta>6/25 and nu>1/20. Put a=delta+6/25, b=nu+1/20 and assume a,b<1. Use separate copies of L_0,L_1,Q,B on the two sides. For a pair of coefficient types, let a',b' be their support exponents and rho,sigma their L2 norm exponents. Logarithmic factors are omitted in the following table, not in the proof: | type on side (D,W)=(x^delta,x^w) | support exponent | norm exponent | |---|---:|---:| | L_0 or L_1 | delta | delta/2 | | Q | delta+w | (delta+w)/2 | | B | delta+w | delta/2+w/4 | The [Bettin–Chandee Theorem 1 and Remark 1](https://arxiv.org/pdf/1502.00769) application already checked in endpoint-fourier.md now has exponents $$ b_1'=\rho+\sigma+{7\over20}(a'+b')+{1\over4}\max(a',b'), \quad b_2'=\rho+\sigma+{3\over8}(a'+b')+{1\over8}\max(a',b'). \tag{9} $$ To check the transfer, on an expanded-divisor box of sizes M,N and a harmonic box of size A, keep the factor (1+Ax/(MN))^(1/2). The divisor norms multiply the Fourier norm O(A^(-1/2)); the two terms before the divisor norms are (MN)^(7/20)(M+N)^(1/4)A^(-3/20) and (MN)^(3/8)(M+N)^(1/8), up to epsilon losses. They increase with M,N, so the global norm bounds and upper supports give (9) on every box. Here b_2'<=b_1' as in the preceding note. Choose fixed tau>0 with b_1'+3tau<1. For MN<=x^(1-tau) the direct discrepancy bound still costs O(x^(1-tau)) times logarithms. On the remaining boxes keep T=ceil(x^(2tau)max(1,MN/x)). The full Vaaler majorant's zero mode is bounded by O(MN/T) times logarithms; its nonzero modes have the same norm bound as the main Fourier terms. The perturbation costs at most x^(3tau/2), and a sufficiently small fixed theorem epsilon costs at most x^(tau/2). Thus (9) gives the same endpoint power saving O(x^(1-tau)log^C x) as before. Splitting gcd=1,2 and partial summation of the explicit log n and log(n-2) weights are unchanged. The zero mode and the sparse norm are both retained; sparsity does not justify deleting the majorant. Write mathfrak b=(17/20)(a+b)+(1/4)max(a,b). Replacing a Q by B decreases the first exponent by w/4. Replacing it by either low term decreases it by at least (17/20)w. It follows that **every sector except QQ has a power saving** whenever $$ \mathfrak b-{\min(6/25,1/20)\over4}<1. \tag{10} $$ This assertion concerns endpoint errors of the expanded expression. It does not say each corresponding raw correlation or density is small. ## 5. One remaining term on an additional rectangle Take delta=277/1000 and nu=467/1000. The expanded exponents are a=b=517/1000. The separate-endpoint QQ first exponent is 20163/20000>1; the largest of all other sector exponents is 19913/20000<1, attained by QB. With tau=1/1000, the latter plus 3tau is 19973/20000<1. These comparisons are checked with exact rationals by [coefficient-structure-validation.js](coefficient-structure-validation.js). Let Delta_{l,j}(x) be the exact CRT count minus density from endpoint-fourier.md (8), zero when gcd(l,j) does not divide 2. Define $$ E_{QQ}(x)=\sum_{\ell,j}Q_{I,V}(\ell)Q_{J,Z}(j) \Delta_{\ell,j}(x). \tag{11} $$ The whole rectangle's density M_{I,J} is O_H(x/log^H x), by the uniform excluded-prime Möbius argument; a,b<1 ensures the cofactor cutoffs are automatic eventually. Summing the other endpoint sectors therefore gives the derived reduction $$ R_{I,J}(x)=E_{QQ}(x)+O_H(x/\log^H x) \quad\hbox{for every fixed }H. \tag{12} $$ Up to this point only the **total** density has been discarded using that proof. The QQ density remains inside Delta in (11). There is also a separate uniform argument for this particular density, as follows. Write C_QQ for the raw correlation obtained by replacing Delta in (11) with the CRT count. Its density M_QQ=C_QQ-E_QQ is exactly $$ M_{QQ}={x\over2}\sum_{p\le V,q\le Z}(\log p)(\log q) \sum_{d\in I,e\in J\atop p\nmid d,q\nmid e} \mu(d)\mu(e){\gcd(dp,eq)\over dp\,eq} 1_{\gcd(dp,eq)\mid2}. \tag{12a} $$ Fix p,e,q and write d=2^epsilon t, t odd, epsilon in {0,1}. If p=2, the condition p not dividing d forces epsilon=0. For odd p, compatibility first requires p coprime to the odd part of eq; the remaining odd d restriction excludes the prime factors of both p and eq. For each allowed epsilon the numerator of the density is constant, at most 2. The d sum is therefore a constant of magnitude at most 2/(epq) times an excluded-prime mu(t)/t sum, with parameter m=2p(eq)_odd and t in (D/2^epsilon,2D/2^epsilon]. Here m<=2V*(2EZ)<=x^2 eventually, since b<1 and V<=x^(6/25). The lower endpoint is at least U/2 and the upper endpoint is below x. The uniform lemma in signed-divisor-grouping.md section 2 applies. After this signed d sum is bounded, the outer sums over e,p,q cost only fixed powers of log x: sum 1/e and sum_{p<=V}(log p)/p suffice, and the restriction q not dividing e can only reduce those absolute costs. Consequently M_QQ=O_H(x/log^H x) for every fixed H. Thus (12) also holds with C_QQ in place of E_QQ. This is an asymptotic density estimate with a written proof; M_QQ is not exactly zero. The exact finite object (11) retains the subtraction. The rectangle has de of order x^(744/1000), outside the old de<=x^0.70 region and disjoint from the previously controlled d~x^0.27 rectangle, eventually. Equation (12) simplifies this further region; the bound on E_QQ is now supplied by [prime-dispersion.md](prime-dispersion.md). No uniform product cutoff is extended. If E_rest is the remainder in endpoint-fourier.md (15), set $$ E_{\rm other}=E_{\rm rest}-(R_{I,J}-M_{I,J}). $$ Then S=C_2x+E_other+E_QQ+O_H(x/log^H x). Both E_other and the necessary joint one-sided improvement remain OPEN. Establishing E_QQ=o(x) alone on this rectangle would not complete a proof of twins. ## 6. Validation and the next arithmetic question The validator checks 172,584 exact log-prime coefficient identities, including 4,403 nonzero nonsquarefree coefficients and 9,718 inputs with two repeated primes that vanish. It checks the support allowance (4) plus the higher-prime-power count, the squarefree witness inequality (7), and all 16 CRT sectors on a full toy interval. There are 1,619 compatible cells, including 595 with gcd=2. At x=4096 in that toy, the total endpoint error is 6.147204950 but the QQ endpoint is -6.294308087; omitting the other sectors is not an exact finite identity. The density is 455.504935205 and is explicitly retained. An independent prime-factor enumeration also reproduces (12a) and the QQ endpoint, including the primitive prime restrictions. Three archived factor prefixes are reused. Their raw QQ/partner values are 0, 0.009877710 and -0.000297641. These are **raw prefix correlations**, not (11): the density has not been subtracted, and neither the full dyadic interval nor an asymptotic rate is measured. Finite norms likewise validate inequalities, not their limiting exponents. The bound output and [retained artifact](data-reuse/coefficient-structure.json) distinguish these quantities and include the input hash. The classification would fail if an allowed prime-power exponent or an interval endpoint were lost. The norm transfer would fail if the smaller norm were claimed without its wider support, or if a density were silently set to zero. Equations (1)–(12) and the finite controls check these points. The analytic imports and asymptotic applications are written arguments, not conclusions of the scripts. The remaining prime structure can be written exactly as $$ E_{QQ}=\sum_{p\le V,\ q\le Z}(\log p)(\log q) \sum_{d\in I,e\in J\atop p\nmid d,\ q\nmid e} \mu(d)\mu(e)\Delta_{dp,eq}(x), \tag{13} $$ where p,q are primes and incompatible CRT pairs contribute zero. The coprimality restrictions make dp and eq squarefree whenever their Möbius factors are nonzero. This identity preserves the prime averages; estimating every p,q separately would reintroduce the norm loss already documented in endpoint-fourier.md. The remaining correlation retains the prime averages in (13). [Endpoint pairing](endpoint-pairing.md) improves the relevant first power budget: on the top expanded boxes, the remaining deficit is 61/20000 near h~MN/x~x^0.034, before strict losses. The lowest frequencies there are already power-saving. The full majorant is controlled by the elementary divisor-count bound in that note's section 5; do not discard it or identify its absolute coefficients with the signed QQ coefficients. [Prime-band completion](prime-band-completion.md) removes the small distinguished primes and low harmonics uniformly, then derives an exact transform with coupled coefficients. [Prime dispersion](prime-dispersion.md) then controls this pilot by regrouping m=dp before Cauchy and retaining the right-prime cross terms; the general full-dual-range operator bound remains insufficient at its proved scope. A one-sided estimate may have a different sufficient budget. The individual squarefree norm reduction is now ruled out at the stated uniform power scope; it is not a reason to discard the phase correlation.