# Full residual coverage and the first-branch moment **The remaining signed endpoint estimate and twin-prime infinitude are OPEN.** This note derives a larger controlled set from the previously specified classical inputs. It gives an exact complement, not a percentage of the problem solved. The finite computation tests the new coefficient handling and set partition; it does not measure an asymptotic saving. The audit changes the next research decision. The exceptional coefficient need not be bounded only by its norm. Retaining its first branch, and including a moment without a distinguished prime, gives estimates for both orientations of the **whole** coefficient. The two resulting edges have different limiting terms. ## 1. Objects, cutoffs and the full domain Use the exact residual and endpoint kernel from [signed-divisor-grouping.md](signed-divisor-grouping.md), including its discrete partial summation and both compatible gcd branches. Put \[ w=6/25,\quad v=1/20,\quad U=V=\lfloor x^w\rfloor, \quad Y=Z=\lfloor x^v\rfloor, \quad D_0=\left\lfloor\frac{x}{V+1}\right\rfloor, \quad E_0=\left\lfloor\frac{x-2}{Z+1}\right\rfloor. \] The exact original divisor domain is \(U=0. Every bound in this section and section 3 is independent of their imaginary parts. Only endpoint sums are twisted; no uniform twisted Mobius mean is asserted. For an interval I contained in (D,2D], the twisted convolution is \[ A_{I,W,1}^{(s)}(\ell)=-\mu(\ell)\ell^{-s}1_I(\ell)\log\ell -\sum_{r\mid\ell,\ 2\le r\le W} \mu(\ell/r)(\ell/r)^{-s}1_I(\ell/r)\Lambda(r). \tag{2} \] It is pointwise O(log ell), uniformly in s. The A0 coefficient is mu(ell)ell^(-s)1_I(ell). On squarefree ell the non-low part of (2) is the first-power sum with original divisor d and weight -mu(d)d^(-s)log p. On a nonsquarefree ell with exactly one repeated prime, ell=p^k m, k>=2, m squarefree, p not dividing m, it is \[ \mu(m)\log p\left[ (pm)^{-s}1_{p^{k-1}\le W}1_I(pm) -m^{-s}1_{p^k\le W}1_I(m)\right]. \tag{3} \] Other nonsquarefree ell give zero. The interval indicators in (3) are disjoint because I has ratio at most two. The first term is the branch previously placed in B_exc. Its original divisor is **pm**, while its expanded divisor is p^k m. This distinction supplies the shorter base interval; it cannot be replaced by a smaller support bound for B_exc. For odd p and a fixed k>=2, split p^k into R<=p^k<2R, set P=R^(1/k), and write m=g d with g in {1,2}. The condition pm in I puts d in O(1) intervals of size D/P. Retain this condition inside the prime sum. The conditions p^(k-1)<=W imply \[ R\ll PW,\qquad P\le W,\qquad D/P\gg1. \tag{4} \] The last assertion uses D>=U and W=V on the left, or E>=Y and W=Z on the right, up to fixed floor constants. The coefficient outside the moment is mu(gd)(gd)^(-s); inside it is p^(-s)log p. Both additional factors have modulus at most one. The extra condition pd in the original interval causes no loss: it restricts the prime set at fixed d and can be dropped after taking a nonnegative bound. Fixed k is chosen **before** Cauchy; there are no mixed-k kernels. Bands in p^k keep every comparison uniform as k grows. Apply [prime-power-dispersion.md](prime-power-dispersion.md) (13), with base length D/P instead of D. Its proof uses only bounded prime weights, so it remains valid for the twists above. If F is the norm of the other coefficient and its support is n~N, the bracket before F becomes \[ \sqrt{Dx/R}+x^{3\tau/2}\left[ D^{5/4}R^{1/2}P^{-1/4} +D^{5/4}R^{1/4}P^{-3/4}+\sqrt{DN/P}\right]. \tag{5} \] There are also x^epsilon and fixed logarithmic factors. Using (4), this is at most \[ \sqrt{Dx}+x^{3\tau/2} \left[D^{5/4}W^{3/4}+D^{5/4}W^{1/4}+\sqrt{DN}\right].\tag{6} \] In particular the first-branch cross budget is smaller than the first-power cross budget by 3w/4 on the left, or 3v/4 on the right. Its zero term still needs an explicit bound; (6) does not supply a uniform extra power saving there. ### Powers of two and the low coefficients The even prime cannot simply use m=g d in every branch. If ell=2^k m with m odd and g=2, the reduced modulus is 2^(k-1)m. For g=1 it is 2^k m. Fix the power before taking the moment. There is only one distinguished prime, so no distinct-prime cross term. For the first branch the base length is comparable to D/2 and the reduced power is at most 2W; for the negative branch it is comparable to D and the reduced power is at most W. A singleton version of the same proof gives \[ F x^{\epsilon}\left[\sqrt{Dx} +x^{3\tau/2}\big(D^{5/4}W^{1/4}+\sqrt{DN}\big)\right].\tag{7} \] Retain the odd-base restriction. The factor theta=2/g is harmless even at p=2: gcd(theta r,c)<=2 gcd(r,c). Thus the full power-of-two gcd is still averaged over the harmonics, and only an absolute factor is added. This is not an assertion that the exact odd-prime gcd identity continues unchanged at p=2. Finally the coefficients A0, the low part of A1 and Q2 have support O(D) and pointwise logarithmic bounds. They admit a **prime-free** version of the moment: take a single multiplier 1, with weight 1 in place of log p. The kernel has modulus d and numerator theta(h1-h2). Its zero-numerator weight is O(1/A); for h1!=h2, the common-divisor average applies. Completion gives \[ F x^{\epsilon}\left[\sqrt{Dx} +x^{3\tau/2}\big(D^{5/4}+\sqrt{DN}\big)\right]. \tag{8} \] One can see (8) directly from Cauchy in n then d: the moment is bounded by f^2[DN/A+x^(3tau+epsilon)(D^(3/2)+N)], where f=min(1,Ax/(DN)). Use f/sqrt(A)<=sqrt(x/(DN)). An arbitrary logarithmically bounded coefficient on d can stand outside this moment; its signs are lost in this Cauchy step. No assertion about cancellation of those signs is needed for (8). The inputs throughout are the already matched [Ramanujan and composite-modulus Weil bounds, Pascadi Lemmas 3.2--3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0), their complete-period term, and the derived nonzero common-divisor gcd average. All harmonic gcd factors and equal-frequency pairs from the prime-power note remain present. Positive and negative harmonic bands are estimated separately. No stronger imported theorem is used. ## 3. Both orientations now include the whole coefficient Against any other coefficient on support n~N with norm F, the largest cross term is still the squarefree first-power term. Equations (6)--(8), the odd negative prime-power terms, and the first-power bound therefore give the uniform endpoint bound \[ F x^{\epsilon}\left[\sqrt{Dx} +x^{3\tau/2}\big(D^{5/4}W^{3/2}+\sqrt{DN}\big)\right],\tag{9} \] up to fixed logarithms. Summing O(log x) powers, power bands and support boxes costs only further logarithms. Same-prime terms are smaller than the displayed cross term. There is no remaining B_exc norm condition. The left orientation uses the reciprocal phase with **both** counted endpoints shifted by 2. The right orientation uses the native phase \[ e\left(-\frac{\theta h\overline m}{n}\right) \left[e\left(\frac{hz_0}{gmn}\right) -e\left(\frac{hz}{gmn}\right)\right]. \tag{10} \] Here m is the left reduced divisor and n the right one. Thus the right moment has negative theta and unshifted z0,z. Absolute completion and gcd estimates are unchanged by that sign. In either orientation the arbitrary other coefficient can have even reduced divisors in the g=2 branch. No unsupported parity restriction is inserted. The cross, zero-numerator and complete-period budgets in (9) are | orientation | cross W | zero J | periods P | |---|---|---|---| | left | 5a/4+b/2+w/4 | 1/2+(a+b)/2-w/2 | b+a/2-w/2 | | right | a/2+5b/4+v/4 | 1/2+(a+b)/2-v/2 | a+b/2-v/2 | Require all three budgets of one orientation to be at most 1-eta, with a fixed eta>0. Choose tau and epsilon sufficiently small compared with eta. Boxes MN<=x^(1-tau) are bounded directly. On the other boxes, the Vaaler polynomial uses T_F=ceil(x^(2tau)max(1,MN/x)). The **entire positive majorant**, including nonzero frequencies, is bounded by [endpoint-pairing.md](endpoint-pairing.md) section 5's divisor-count argument. It is not multiplied by a paired factor. Twists preserve its pointwise coefficient bounds. Uniformity in z restores the original logarithmic endpoint weights by partial summation. Consequently the twisted endpoint sum is O(x^(1-c_eta)) for some c_eta>0, with logarithms absorbed, uniformly in both imaginary parts. This is stronger than any fixed logarithmic saving, but no effective onset is asserted. On the limiting domain (1) the sufficient conditions simplify exactly: \[ \boxed{\quad \delta+\nu<19/25 \quad\hbox{or}\quad 5\delta+2\nu<123/50.\quad} \tag{11} \] For the right orientation set theta=delta+nu. Then \[ J_R=\frac{31}{50}+\frac\theta2,\qquad W_R\le\frac54\theta+\frac3{200},\qquad P_R\le\theta+\frac{43}{200}. \tag{12} \] At theta=19/25 the latter two upper budgets are 193/200 and 39/40, strictly below one; J_R reaches one. Conversely J_R<1 requires theta<19/25. The left cross budget is W_L=(5delta+2nu)/4+77/200. As in the preceding note, W_L<1 and delta>=w imply J_L<24/25 and P_L<4/5. This proves both simplifications, with uniform slack for every compact subset of the strict inequalities. The generic paired region is contained here: C<1 implies a+b<34/33<21/20, which places it in the right region. The earlier dispersion regions have W_L<1 and are contained in the left region. Thus the audit does not discard an earlier controlled corner. The frontier lines meet at (delta,nu)=(47/150,67/150). The product-exponent supremum of their union is 87/100, approached at (6/25,63/100) along the left edge. This follows from delta+nu<123/100-3delta/2<=87/100. It remains a regional supremum. The **uniform** product threshold from (11) is every fixed exponent below 19/25, not 87/100. ## 4. Separate the coupled cuts with a uniform cost Fix theta<19/25 and lambda<123/50, independent of x. Define integer cuts \[ L_\theta=\lfloor x^\theta\rfloor,\qquad K_\lambda=\lfloor x^\lambda\rfloor,\qquad \mathcal A=\{de\le L_\theta\},\quad \mathcal B=\{d^5e^2\le K_\lambda\}. \tag{13} \] Intersect these with the exact full domain in section 1. On a dyadic box meeting A, its lower endpoints satisfy DE=1, c=1/log x and T_P=x^10, write \[ P_K(q)=\frac1{2\pi i}\int_{c-iT_P}^{c+iT_P} \frac{(K+1/2)^s q^{-s}}s\,ds. \] Then \[ 1_{q\le K}=P_K(q)+O((K+1)/T_P), \tag{14} \] uniformly for the integers here. To verify the formula, put u=log((K+1/2)/q). Fourier inversion of exp(-c u)1_(u>0), or the elementary one-pole contour integral, gives the untruncated integral 1_(u>0). Integrating each tail by parts bounds its error by O(exp(cu)/(T_P |u|)). Integer spacing gives |u|>>1/(K+1); exp(cu)<=exp(c log(K+1/2))=O(1) for the fixed polynomial cuts. The half-integer shift excludes equality at the jump. For q=de the integrand is a product of twists d^(-s), e^(-s). For q=d^5 e^2 it is d^(-5s)e^(-2s). Their real parts are nonnegative, so sections 2--3 apply uniformly for all imaginary parts in the integral. The integral of 1/|s| costs O(log x). For A intersect B use both integrals: the twists are d^(-s-5t), e^(-s-2t), costing O(log^2 x). Both budgets are available on a box meeting that intersection; one suffices. There is no height-dependent derivative loss because the twists belong to the divisor coefficients, not the counted endpoint. The sum of absolute values of the unaggregated endpoint terms on the whole domain is O(x^2 log^C x): each discrepancy with its logarithmic weights is O(log^C x), there are at most D0 E0 original divisor pairs, and at most VZ weighted expansion terms up to logarithms. Equation (14) therefore has total error O(x^(2+max(theta,lambda)-10)log^C x). For the double cut use |P_K|<=1+O((K+1)/T_P), which also bounds the product approximation error. These errors are negligible. This is a justified cost for the coupled cuts, not an appeal to a rectangle estimate on a nonrectangular set. Finally, bound each **untwisted density** before this separation. At fixed e the allowable d form an interval: its upper bound is L_theta/e, (K_lambda/e^2)^(1/5), or their minimum for the intersection. For the union it is their maximum. Clip with U,D0. The uniform excluded-prime Mobius lemma in signed-divisor-grouping.md applies on these intervals; the remaining harmonic sums cost fixed logarithms. It gives O_H(x/log^H x) for each density and hence for their complement by subtraction from the full density. No large-height twisted Mobius estimate has been used. Combining the endpoint and density bounds proves, for each fixed H, \[ R_{\mathcal A},\ R_{\mathcal B},\ R_{\mathcal A\cap\mathcal B},\ R_{\mathcal A\cup\mathcal B}=O_H(x/\log^H x). \tag{15} \] This improves the preceding elementary uniform range de<=x^(7/10). That argument also permitted any fixed exponent below 71/100; neither fact is a claim that its endpoint or this note's endpoint is included. ## 5. The exact remaining sum and sufficient consumer For a concrete fixed choice take theta=3/4 and lambda=49/20. The right zero budget is at most 199/200 and the left cross budget at most 399/400, before arbitrarily small truncation and epsilon losses. Define \[ \mathcal W_* =\{U\lfloor x^{3/4}\rfloor, \ d^5 e^2>\lfloor x^{49/20}\rfloor\}, \] \[ E_*(x)=\sum_{(d,e)\in\mathcal W_*}\mu(d)\mu(e) \sum_{r\in\mathcal P_V,\ s\in\mathcal P_Z} \alpha_r\alpha_s\mathcal E_{d,e,r,s}(x). \tag{16} \] Every symbol in this kernel is the exact weighted, compatible CRT kernel in signed-divisor-grouping.md (17)--(18), including zero for incompatible gcd and the original endpoint convention. Thus no boundary contribution is implicit in (16). Inclusion-exclusion accounts for overlap of the two controlled cuts exactly. The quantitative reduction in [endpoint-target-audit.md](endpoint-target-audit.md) now becomes \[ \sum_{x/20). \tag{17} \] The sufficient **OPEN** hypothesis is still, for fixed c,K>0, C2*x+E_*(x)>=c*x/log^K x on an unbounded set of dyadic scales, or the precise logarithmically rescaled positive-average condition from that note with E_* substituted. The union bound (15) justifies this substitution at every fixed logarithmic precision. No such margin has been established. The limiting unhandled region of this family is explicitly \[ \delta+\nu\ge19/25,\qquad5\delta+2\nu\ge123/50, \quad\hbox{inside (1)}. \tag{18} \] For the concrete fixed cuts in (16), retain the strips between 3/4 and 19/25, and between 49/20 and 123/50. They are not discarded as negligible. For example (delta,nu)=(2/5,2/5) lies in the open complement even of the limiting region. Its right cross/zero/period budgets are respectively 179/200, 51/50, 21/25; the left cross budget is 217/200. At this box, improving the already smaller right cross term does not close the bound. There is no argument converting the geometric area of (18) into the size or sign of E_*. ## 6. Next arithmetic target: retain the divisor signs before the extra Cauchy For the full product edge the limiting term is J_R, the equal-frequency part after the second Cauchy inequality. For the left edge it is W_L, the nonzero first-power cross term. The exceptional norm is no longer the term that must be improved at either edge of (11). A specific next test is the right first-power sector at delta=nu=2/5, with a=16/25 and b=9/20. In the native orientation set \[ Y_e(m)=\sum_{\substack{q\ { m odd\ prime},\ q\le Z\\ eq\sim N,\ q\nmid e}} \sum_{h\sim A}(\log q)c_h1_{(m,eq)=1} e_{eq}(-\theta h\overline m) \left[e\left(\frac{hz_0}{gmeq}\right) -e\left(\frac{hz}{gmeq}\right)\right], \] with ge in the original right interval and g in {1,2}. Fix prime and harmonic bands as in the existing proof, retaining the exact support cuts. Write gamma_e=mu(ge). First Cauchy, in the other divisor, requires an upper bound on \[ \mathcal M=\sum_{m\sim M}\left|\sum_{ge\in J} \gamma_eY_e(m)\right|^2. \tag{19} \] The current second Cauchy replaces it by O(E) sum_(ge in J,m~M)|Y_e(m)|^2. Instead expand (19) as its e1=e2 part plus the real signed sum \[ \mathcal X=\sum_{\substack{ge_1,ge_2\in J\\e_1\ne e_2}} \mu(ge_1)\mu(ge_2) \sum_{m\sim M}Y_{e_1}(m)\overline{Y_{e_2}(m)}. \tag{20} \] This retains Mobius signs absent from the earlier moment. Its complete moduli are lcm(e1*q1,e2*q2); the common factors and rational-frequency collisions must be computed before completion. The zero-frequency part within a fixed e is nonnegative; cancelling it after taking an absolute second moment is not an available argument. For this **specified sector and box**, the e-diagonal in (19) has the old three block budgets reduced by nu/2=1/5: 139/200, 41/50, 16/25. This follows by removing the factor E in the squared bound, not by assuming additional cancellation. They are all below one. Since the other coefficient has F^2<0 would suffice for this sector, after the diagonal is included. The existing Cauchy bound has moment budget x^(7/5+epsilon), so this target asks for more than a 1/25 saving in its exponent. Keep endpoint uniformity, power/prime bands and logarithmic losses in this comparison. This target is **OPEN** and is not yet a full-box result: the other coefficient sectors must also be handled, and the full complement (18) is larger than this one box. The [structural literature audit](structural-literature-audit.md) derives the exact cross-divisor kernels in (20), including equal rational frequencies, and checks relevant theorem interfaces. The [grouped-divisor moment](grouped-divisor-moment.md) now controls this entire benchmark and an additional exact cut by retaining the full coefficient before a further Cauchy inequality. It specifies E_dagger as the current complement and a remaining small-common-divisor target. The global endpoint margin remains OPEN. Improving W_L alone remains a possible local extension, with a different coverage consequence. ## 7. Validation, falsifiers and limits [residual-coverage-validation.js](residual-coverage-validation.js) tests the twisted decomposition against the original convolution, including clipped intervals and the p=2,g=2 reduction. It constructs first-branch moments from the original divisors and independently regroups them by the shorter base, checking both orientations. Exact BigInt residues test the endpoint shift, with an active wrong-shift control. Rational budget checks test the two simplified regions and prior-region inclusion. The saved [factor windows](data-reuse/factor-windows.json) are reused for the original signed beta weights and exact inclusion-exclusion masks. The monomial cut is evaluated with BigInt, including values above the safe integer range. Nonzero overlap makes omitting the intersection an active failed control. Full inputs and summaries are retained in [data-reuse/residual-coverage.json](data-reuse/residual-coverage.json). The validator completed in 1.4 seconds: 146,016 twisted coefficient identities, 47,172 exact phase residues, forty first-branch/low moment checks with 535,900 expanded pairs, and 117,711 rational grid checks. The archived partition uses 273,177 weighted divisor pairs; 231,071 monomials exceed the safe integer range. These counts describe finite checks only. The computation is bounded by three minutes, within the user's allowance. All fourteen strict QC gates report zero findings. The verifier self-tests pass all 58 known positives and 47 controls. The independent existing numerical audit passes 251/251 checks in 164.7 seconds. The new validator and that audit together take under three minutes of numerical runs. A wrong first-branch power, dropped p=2 reduction, unshifted reciprocal endpoint, or double-counted intersection would invalidate the extension; the finite identity checks for these have run. They do not prove the analytic uniformity, which is supplied by the written bounds and the explicit separation argument. Mechanical repository gates do not act as independent mathematical refereeing. No novelty, optimality, effective onset or asymptotic twin lower bound is claimed.