# A signed transfer for a growing range of small cofactors **The sufficient twin-prime margin remains OPEN.** This attempt obtains a signed estimate for a larger subfamily of the corner, importing a theorem from the primary preprint cited below. It does not estimate the whole corner or claim novelty in the literature. The derivation below, its finite falsifiers and the failed extension are the outcome; a new representation alone would not have supplied an estimate. Starting checkout: `8edecf0`, clean. Read before this attempt: [the corrected review](transition-round-audit.md), the transition reports, [prime-band-transfer.md](prime-band-transfer.md), the relevant question and outcome entries, and the owning sharp and smoothed coefficient arguments. The earlier transfer covers the prime-cofactor case s=t=1. This attempt retains additional cofactor divisibility instead of treating mu(n/s) as mu(n) or applying a theorem directly to a nonmultiplicative convolution. ## 1. The estimate and its scope Let x=2N, J_x=(N,2N], and fix 0=z_i. Use any of these three real-argument profiles: \[ v_i(u)=1_{u>D_i},\qquad v_i(u)=\tau_i(u)=\rho_i(u)1_{D_i=1 define the prime-r coefficient restricted by the remaining cofactor s: \[ A_{i,H}^{v}(m)= \sum_{\substack{spd=m\\p>W_i\ {\rm prime}\\s\le H}} \mu(d)\log p\,v_i(d). \tag{1} \] This is an actual subfamily, with multiplicity, of the full coefficient. The input m need not be squarefree; no coprimality with s is assumed. For the sharp profile write C_(i,H); for transition and smooth profiles write T_(i,H) and Ctilde_(i,H). Pointwise C_(i,H)=T_(i,H)+Ctilde_(i,H). Thus estimating the sharp product already estimates the **joint sum** of its four restricted transition pieces. **Derived transfer, using the imports in section 3.** There exist positive absolute kappa,d such that, with H(x)=floor((log x)^kappa), for any fixed choice of the two profiles above, \[ \frac1{\log X}\sum_{\sqrt X\le 2^j\le X} \frac{\left|\sum_{2^j=1 define a multiplicative function F_a by F_a(1)=1. If p does not divide a, let F_a(p^v)=mu(p^v). If p^e exactly divides a, put \[ F_a(p^v)= \begin{cases} 1,&v=e,\\ -1,&v=e+1,\\ 0,&v\ge1,\ v\notin\{e,e+1\}. \end{cases} \tag{3} \] Then |F_a|<=1, and, **on multiples of a**, \[ F_a(n)=\mu(n/a). \tag{4} \] This follows prime by prime, including the powers of primes dividing a. For example F_4(4)=1 and F_4(8)=-1, although mu(4)=mu(8)=0. Away from multiples of a, (4) is not asserted: F_a(1)=1 is already a counterexample. Replace mu(m/p) by -mu(m) in (1), with m=n/s, only after accounting for its failure when p^2 divides m. Define \[ B_{i,H}^{v}(n)=-\sum_{\substack{s\le H\\s\mid n}}\mu(n/s) \sum_{\substack{p\mid n/s\\p>W_i}}\log p\, v_i\!\left(\frac{n}{sp}\right). \tag{5} \] Off integers divisible by the square of a prime above W_i, (1) and (5) are equal: mu(m/p)=-mu(m) whenever p divides m but p^2 does not, even if m has a square at another prime. Both coefficients have pointwise bound at most a constant times the divisor function times log x. The exceptional support has size at most x sum_(p>W_i)p^-2 << x/W_i. Consequently, for every fixed epsilon>0, \[ \sum_{n\in J_x} A_{L,H}^{v}(n)A_{R,H}^{v'}(n-2) =\sum_{n\in J_x} B_{L,H}^{v}(n)B_{R,H}^{v'}(n-2) +O_\epsilon(x^{19/20+\epsilon}). \tag{6} \] The bound is uniform in H; it uses the divisor bound, not an H-dependent pointwise estimate. Take epsilon<1/20. This error concerns repeated large primes in n, and is distinct from the already negligible proper-prime-power r branch of the full coefficient. For fixed s,t the two divisibility conditions are compatible precisely when g=gcd(s,t) divides 2. If compatible, they give one residue b_(s,t) modulo \[ Q=\operatorname{lcm}(s,t)=st/g. \tag{7} \] On that progression the two Mobius factors in (5) are F_s(n) and F_t(n-2). This is the coefficient class to which the source is applied. The progression, including its density, is part of the application. ## 3. The source match, with its normalization retained Primary statements checked on 2026-09-06: [Tao–Teravainen, arXiv:2512.01739v2, Theorem 3.1(ii), (3.3)–(3.4)](https://arxiv.org/html/2512.01739v2) and [MRT, arXiv:1503.05121v3, (1.12)](https://arxiv.org/html/1503.05121v3). Their proofs are imported, not reproduced. The equidistributed alternative and its extra prime-value condition are not used. For the former theorem take its parameter mathcal L=(log X)^(1/4). For some sufficiently small absolute c>0 its resulting rate is (log X)^(-c), its allowed progression modulus is Q<= (log X)^c, and its exceptional N-set in [sqrt X,X] has logarithmic measure O(log X (log X)^(-c)). The correlation is normalized by **Q/N**. Use h_1=0, h_2=-2 and the representative 0<=b_(s,t)= (7/24)log log X-O(log log log X). It exceeds (1/4)log log X by an unbounded amount. This checks the non-pretentious source hypothesis uniformly for the whole family. For any fixed weight pair a,a', inserting (9), applying the source on the progression (7), using the trivial AP count on its exceptional set, and then Tonelli gives \[ \frac1{\log X}\int_{\sqrt X}^{X} \frac{\left|\sum_{NW_i(y)}. \tag{14} \] The profile subscript includes D_i(y),z_i(y) and their exact floors. The weights satisfy the hypotheses of section 3. One may use, for example, alpha=1/100 and beta=1/3 uniformly over all cells, both sides and s<=H_X, for sufficiently large X. The two sides' prime ranges are disjoint, and every active prime exceeds H_X. There are two comparisons: replacing m=n or n-2 by x=2N in the profile, then replacing x and its cuts by y. A profile jump can only change primes in intervals whose log endpoint widths are O_eta(delta log X+1). Away from the jumps, rho and h have log-argument slope at most 1/L_i; variation of the denominator L_i as x changes has the same O_eta(delta+1/log X) cost. Hence there are nonnegative, n-independent prime-weight envelopes for the differences with \[ \sum_p\frac{|\Delta a_i(p)|}{p}\ll_\eta\delta\log X+1, \qquad \sum_p\frac{a_i(p)}p\ll_\eta\log X. \tag{15} \] To obtain (15), use sum_(p<=u)log p/p=log u+O(1) at the interval endpoints. Floors are covered by the O(1) widths. The shifted lower endpoint n-2>N-2 changes the comparison by a bounded factor as well. The strict lower divisor cut in tau is included among the jumps. For fixed s,t and primes p,q in the respective ranges, \[ \#\{NH_X and p!=q; without these facts gcd(sp,tq)=gcd(s,t) would be false. After (15), (16) and (12), the absolute normalized product error is \[ O_\eta\!\left((1+\log H_X)^2 (\delta+1/\log X)+X^{-1/10}\right). \tag{17} \] For the O(1) CRT errors, a crude count suffices: at most H_X^2 times O(X^(29/100+4eta+o(1))) prime pairs, each with weight O(log^2 X). Since N>=sqrt X and eta<1/400, division by N log^2 X leaves a fixed power saving; polylogarithmic factors fit inside X^-1/10. This also prices the profile products and exact endpoint rounding. ## 6. Finish the transfer, then attempt the full consumer For each frozen cell, the dyadic estimate from section 4 can be bounded by its nonnegative sum over the whole outer scale range. The mesh costs O(1/delta). Thus the left side of (2), first with B in place of A, is at most \[ O_\eta\!\left((1+\log H_X)^2 \{\delta^{-1}(\log X)^{-c/2}+\delta+1/\log X\} +X^{-1/10}\right). \tag{18} \] Choose delta=(log X)^(-c/4), decreasing c below 1 if necessary. The answer is O_eta((log log X)^2 (log X)^(-c/4)). Absorb the iterated logarithm by setting d=c/8>0. The repeated-large-prime error (6) is smaller than any fixed logarithmic saving. This proves (2) from the named imports. The same calculation before sampling gives a continuous average with exponent c/2, up to the iterated-logarithm factor. The extension covers s>1, t>1 and nonsquarefree n that the earlier s=t=1 transfer did not represent. For the sharp choice it bounds the joint restricted corner, so it is not merely a reformulation or a separate norm bound. It also applies to each restricted transition or smoothed profile and their mixed products. **The attempt to pass to the full corner stops at two priced steps.** The full prime-r cofactor range reaches x/D_iW_i=x^(2eta+o(1)). At that range (s,t)|2 forces Q=st/(s,t), which can be a power of x, outside the source's polylogarithmic modulus range. Even granting a hypothetical uniform modulus extension with the same rate, (12) would cost O_eta(log^2 X), rather than O((log log X)^2). This triangle summation would then supply only an x log^(4-c)-type budget before mesh losses, insufficient even to improve the existing x log^2 x norm budget when c is small. Neither failure excludes a method that retains cancellation between the cofactor pairs or uses a different theorem. For the exact consumer let C_i^P be the full prime-r sharp coefficient, U_i=C_i^P-C_(i,H), K_H=sum_J C_(L,H)(n)C_(R,H)(n-2), and put \[ J_H=\sum_{n\in J_x} \{C_{L,H}(n)U_R(n-2)+U_L(n)C_{R,H}(n-2)+U_L(n)U_R(n-2)\}. \] The existing proper-power estimate gives, with any fixed H_0>0, \[ E_\dagger=E_{\rm out}+K_H+J_H+O_{H_0}(x/\log^{H_0}x). \tag{19} \] Equation (2) does not make K_H negligible at scale x: it gives an averaged upper budget growing like log^(2-d) X for |K_H|/N. J_H includes both mixed tails and is unestimated; E_out is also unestimated at the required one-sided precision. No common-scale positive margin follows. The subsequent [global cutoff assessment](global-cutoff-averaging.md) refines the research priority: initial cutoffs can be averaged in the whole identity, representing the full residual with smoother coefficients. The existing norms force same-input cancellation between the corner and its complement. This gives a reason to test the complete global pair before extending individual cofactor terms. No shifted estimate follows. ## 7. Falsifiers, records and the next obligation [cofactor-progression-transfer-validation.js](cofactor-progression-transfer-validation.js) checks (3)–(7), the cofactor CRT branches, actual sharp/transition/smooth coefficient reconstructions, the exact split and the mixed-tail identity. Controls exercise nonsquarefree inputs, failure away from s|n, prime bands meeting the cofactor, repeated large prime squares of both signs, missing mixed tails, and the Q/N sampling endpoint error. Sharp identities use integer arithmetic for Mobius values; weighted numerical identities use the stated floating tolerance. These are finite checks, not evidence for the imported theorem or the asymptotic rate. A wrong source normalization, loss of uniform non-pretentiousness in s or frequency, an unpriced profile jump, or a CRT error of size N would break this transfer. Their bounds and hypotheses are written above and checked at the source where imported. No large census was run. **Outcome:** derived signed scale-average control for a growing small cofactor subfamily; failed extension to the full signed margin. The current first priority is the complete pair in global-cutoff-averaging, which retains cancellation with the coefficient complement. If pursuing the present cofactor decomposition, J_H and its mixed terms still need an estimate, and a faster single-pair rate must be priced against the full cofactor cost. Repeating the same termwise modulus-one lift or calling (2) an o(x) estimate would not advance that question. Repository verification on 2026-09-06: the linked finite validator passed; `node research/qc.js --index --strict` reported zero findings; `node research/qc/selftest.js` retained all 58 positive and 47 negative controls; `node research/audit-numbers.js` passed all 251 checks; `git diff --check` was clean. These gates check finite identities, recorded numbers and document consistency. The analytic derivation is a direct attempt with a self-audit, not an independent mathematical review.