# Reversing the dispersion and averaging the gcd loss **The complete endpoint remainder and twin-prime infinitude remain OPEN.** This note derives a larger region of controlled full rectangles. Its new ingredients relative to [prime-dispersion.md](prime-dispersion.md) are a different orientation of the same second moment and an elementary average of its gcd loss. Finite algebra and rational arithmetic checks are separate from the proof using classical inputs. No novelty or effective onset is claimed. ## 1. Statement of the region Keep U=V=floor(x^(6/25)), Y=Z=floor(x^(1/20)), and let D=floor(x^delta), E=floor(x^nu), I=(D,2D], J=(E,2E]. Set $$ w=6/25,\quad v=1/20,\quad a=\delta+w,\quad b=\nu+v, \quad t=a+b,\quad u=\max(0,t-1), $$ $$ C(a,b)=3/20+(7/10)(a+b)+(1/4)\max(a,b). \tag{1} $$ Assume delta>w, nu>v, a,b<1 and the following strict inequalities: $$ \begin{split} C(a,b)&<1+w/4=53/50, &\text{left non-Q sectors},\\ a+b-w/2&<1, &\text{identical-pair term},\\ (5/4)a+(1/2)b+w/4&<1, &\text{cross-prime Weil term},\\ C(a-w+u,b)&<1. &\text{small left primes} \end{split} \tag{2} $$ Then for every fixed H, $$ R_{I,J}(x)=O_H(x/\log^H x). \tag{3} $$ The endpoint error has a fixed power saving; the density has arbitrary fixed logarithmic savings as in [endpoint-fourier.md](endpoint-fourier.md). Constants depend on the fixed parameters and their strict margins. Equation (2) is a sufficient region, not an optimality statement. The previously established region C(a,b)<1 remains available separately. ## 2. Remove only the left low and nonsquarefree sectors Use L_0,L_1,Q,B from [coefficient-structure.md](coefficient-structure.md). Leave the entire right coefficient in place: it may be any one of its four types, or A_0,A_1 before that split, each logarithmically bounded and supported up to 2EZ. Only the left Q coefficient needs dispersion. Here is the sparse-norm transfer for the **paired** bound. If the support exponents are a',b' and the norm exponents rho,sigma, its two powers are $$ C'=3/20+\rho+\sigma+(a'+b')/5+\max(a',b')/4, \quad G'=\rho+\sigma+3(a'+b')/8+\max(a',b')/8. \tag{4} $$ To check this rather than assume a norm discount, on a divisor box M,N multiply the bound in endpoint-pairing.md (9) by the actual norm product divided by sqrt(MN). The resulting monomials are x^(3/20)||xi||||zeta||(MN)^(1/5)(M+N)^(1/4) and ||xi||||zeta||(MN)^(3/8)(M+N)^(1/8). Both increase with M,N. Endpoint integration divides each coefficient by its own divisor and preserves separation. The same truncation and full majorant argument therefore give (4), with arbitrarily small fixed losses. Boxes MN<=x^(1-tau) still use the pointwise logarithmic coefficient bounds. This uses [Bettin--Chandee, Theorem 1 and Remark 1](https://arxiv.org/pdf/1502.00769), in the already checked convention; no new bilinear theorem is imported. On the left B sector, a'=a and rho=a/2-w/4, so the first exponent is C(a,b)-w/4. For either left L sector it decreases by at least 7w/10. Thus the first line of (2) controls all three left non-Q types against every right type. The second term is also below 1: writing G=7(a+b)/8+max(a,b)/8, one has G<=(25/22)(C-3/20). Consequently G-w/4<2143/2200<1 under C<53/50; either L sector saves at least 7w/8 instead. Small tau handles all strict losses. The uniform endpoint estimates permit partial summation of the original weights 1, log k, log(k-2), log k log(k-2). These logs refer to the original counted integer k, not the expanded divisor variables below. There is no requirement to dispose of the right nonsquarefree sector by its smaller norm: it will be allowed as an arbitrary coefficient. ## 3. Cut the small left primes and keep every harmonic By the second line of (2), u=1. No uniform low-h cutoff or frequency-overlap condition is needed in this orientation. ## 4. Reciprocity keeps a required shift of both endpoints For (m,n)=1, inverse reciprocity gives inv(m)/n+inv(n)/m = 1/(mn) modulo 1. Put theta=2/g, g in {1,2}. The original positive-h phase and endpoint factor therefore satisfy $$ \begin{split} &\mathrm e\left(-\theta h\frac{\overline m}{n}\right) \left[\mathrm e\left(\frac{hz_0}{gmn}\right) -\mathrm e\left(\frac{hz}{gmn}\right)\right]\\ &=\mathrm e\left(\theta h\frac{\overline n}{m}\right) \left[\mathrm e\left(\frac{h(z_0-2)}{gmn}\right) -\mathrm e\left(\frac{h(z-2)}{gmn}\right)\right]. \end{split} \tag{7} $$ Here z_0=x/2 and z_0<=z<=x. Dropping the shift by 2 would change the sum. The endpoint derivatives still have size O(Ax/(MN)) on the box. Write m=dp, gd in I. Since p is odd, the exact left Q weight is -mu(gd)log p with p not dividing d. Let beta_g(n) be the actual right coefficient at gn; it is bounded by a fixed power of log x and its norm on n~N is O(sqrt(N) log^C x). Define P_d to contain the retained primes with dp~M and p not dividing d, and define $$ \begin{split} \widetilde\Phi_{h,p,d}(n)&= \mathrm e\left(\frac{h(z_0-2)}{gdpn}\right) -\mathrm e\left(\frac{h(z-2)}{gdpn}\right),\\ Y_d(n)&=\sum_{p\in P_d}\sum_{h\sim A}(\log p)c_h 1_{(n,dp)=1} \mathrm e_{dp}(\theta h\overline n) \widetilde\Phi_{h,p,d}(n). \end{split} \tag{8} $$ For g=2 the original left Mobius weight forces odd d, but a right nonsquarefree coefficient can have even n. No odd-n restriction is imposed: for example the validator includes B_(J,13)(132)=-log 2, corresponding to g=2, n=66 and J=(24,48]. The block is exactly -sum_(gd in I) mu(gd) sum_(n~N) beta_g(n)Y_d(n). First Cauchy in n removes beta_g, then Cauchy in d gives $$ |\text{block}|^2\ll ND\log^C x\sum_{gd\in I}\sum_{n\sim N}|Y_d(n)|^2. \tag{9} $$ Enlarging the nonnegative second moment to all d is allowed; squarefree and parity restrictions from mu(gd) were not imposed on the enlarged sum. Both gcd branches are thus covered. The condition p not dividing d and the (n,dp)=1 factors remain inside Y_d. ## 5. Average the gcd loss before taking its worst value Expanding (9), the n kernel has modulus c=dp for equal p, and c=dp_1p_2 for distinct primes, with numerator respectively theta(h_1-h_2) or theta(h_1p_2-h_2p_1). All n must be units modulo c. Identical pairs alone have numerator zero. On every other pair its numerator r is nonzero, polynomially bounded in x, coprime to the distinguished prime or primes, and $$ \gcd(r,c)=\gcd(r,d). \tag{10} $$ The prime cut p>2h proves these assertions even for theta=2. When d is summed, its allowed set includes the dp_i box restrictions and p_i not dividing d. These may be dropped after taking a nonnegative upper bound, since they define a subset of d of order D. For r!=0 and 0=1. Sum over divisors of |r|. The translated gcd=2 branch has D_0=D/2 and costs the same order. No assumption r=1 and N<0. Thus (2) leaves a strict margin on every term. Choose tau and then the theorem epsilons small enough for all margins, the prime cut (5), the small-box bound and the full-majorant bound. Summing the logarithmically many boxes and blocks preserves a fixed power saving. Restore the original smooth weights by partial summation and consume the total density estimate. This proves (3). On balanced expanded supports a=b=s, (2) gives the particularly simple sufficient family $$ 12/25x^.12, so p>2T eventually. The dispersion polynomial is O(x^.9999) after its losses; the other components retain their own power savings. Equation (3) controls the full rectangle at product scale x^.784, compared with x^.744 for the previous pilot. This does not bound every pair up to that product or measure a fraction of the full residual. For a falsifying scope control take s=269/500=.538. The cross-prime budget becomes 2003/2000=1.0015 while the other displayed conditions still have slack. This bound is insufficient there; it does not prove that cancellation fails. Likewise (15) is a strict boundary of this sufficient balanced calculation, not a universal barrier. The initial attempt to extend the previous orientation by a single low-h cutoff has its own narrower budget: at balanced s it requires 4s-41/20a, and the diagonal and prime-cut conditions still have slack with kappa=.12; decreasing both coordinates slightly satisfies every strict condition. Consequently the exact supremum of original product exponents in this sufficient region is $$ \sup(\delta+\nu)=5397/6700\simeq .805522. \tag{16} $$ This optimizes **the inequalities (2)** only. It neither excludes another estimate past that value nor asserts control of every pair with product below it. Near this corner both the left B budget and the cross-prime budget become tight; improving only the balanced boundary misses that second constraint. ## 7. Validation and next work [dispersion-range-validation.js](dispersion-range-validation.js) and [its retained artifact](data-reuse/dispersion-range.json) check the shifted reciprocity identity, exact moments with the actual right A coefficients, both gcd branches, nonunit restrictions, the averaged-gcd inequality and rational region budgets. Controls include nonidentical prime pairs with nontrivial gcd loss, omission of the endpoint shift, and the excluded r=0 case. Saved factor inputs supply the prime table; no larger sieve is run. The right-coefficient probe is A_1+(2+i)A_0, an allowed complex combination of the actual coefficients that exercises all four sector types. It is an algebra test for the general coefficient bound, not a measurement of R. These finite checks can falsify algebra and bookkeeping, not prove the classical asymptotic inputs. Missing the shift, deleting a right sector, using (11) on r=0, or averaging after the wrong absolute-value step would invalidate the result; the proof keeps each explicitly. The retained run has 5,472 gcd inequalities, 127,068 shifted endpoint identities, twelve full weighted moments and 693,684 pair-phase checks. It includes all four right coefficient types and 199,400 nonidentical terms with nontrivial gcd. Exact rational checks caught a period-term addition using the original right-divisor exponent instead of b; the corrected balanced value is .6855 and the leading budgets are unchanged. All fourteen strict QC gates pass with zero findings, the verifier self-tests pass (58 positives, 47 controls), and the existing numerical audit passes 251/251 checks. These are distinct from reviewing the proof. The [sparse-dispersion follow-up](sparse-dispersion.md) now controls left B against right Q using the opposite orientation, including the frequency collisions that arise when its harmonics exceed its right primes. It retains the sparse norm and pairs endpoints inside the moment. Combining that estimate with BB's two sparse norms relaxes the first inequality here and gives a larger region, with product-exponent supremum 1368/1675. The follow-up also writes the exact signed cross-prime moment before section 5's final triangle inequality and specifies its one-sided upper consumer. Its remaining tight budgets are BB and the left-prime cross moment. Consume those calculations before proposing another orientation or frequency cut. The full residual outside the controlled rectangles remains open; no global coverage or twin lower bound follows.