# Joint small folds cancel; the signed reconstruction remains open Internal continuation, 2026-09-05. **No new twin-prime lower bound follows.** This executes the growing-depth and reconstruction check suggested by [bilinear-fold-attack.md](bilinear-fold-attack.md). All a,b,w,β,U,V,B below are defined in [prime-detection-spec.md](prime-detection-spec.md). Finite CRT identities are checked in [prime-detection-validation.js](prime-detection-validation.js). The subsequent [shifted-prime decomposition](shifted-prime-decomposition.md) controls the second Type I pieces of the actual sum; its residual estimate remains OPEN. ## 1. The joint local model Fix L>0. Let Q be even and squarefree, with Q≤(log x)^L; Q may depend on x. For odd q|Q use A_q,b_q from the single-fold note; set A₂(t)=b₂(t)=2·1_(t odd). Define $$ A_Q(t)=\prod_{q\mid Q}A_q(t) ={Q\over\phi(Q)}1_{(t-2,Q)=1},\qquad b_Q(t)=\prod_{q\mid Q}b_q(t),\qquad r_Q(t)=A_Q(t)-b_Q(t). \tag{1} $$ These are explicit periodic local factors. They are not the functions Λ(t−2) and b(t). In particular, A_Q is a normalized roughness indicator, not a prime indicator. Write C_Q=∏_{q|Q,q>2}(1−1/(q−1)²); then b_Q(t)=2C_Q·1_(t odd)·f_R(t), where R is the set of odd prime divisors of Q. For every integer d and every g|Q, $$ \sum_{k\bmod Q\atop(k,Q)=g}r_Q(dk)=0. \tag{2} $$ Proof: the gcd class is a product of zero/nonzero residue classes by CRT. At an odd q dividing d or g, both factors take the same constant value. At an odd q dividing neither, multiplication by d permutes the nonzero residues, on which the average of A_q is b_q. At 2 the two factors agree pointwise. Average the products coordinate by coordinate. ∎ Thus the single-fold conditional-mean identity survives joint folds exactly. It does not require independent primality assumptions. ## 2. The specific beta weights balance uniformly For V≤N≤x and any interval K⊂[1,N], put $$ W_a(K)=\sum_{k\in K\atop k\equiv a\pmod Q}\beta_V(k). $$ Uniformly for a,b with (a,Q)=(b,Q), every fixed A,L>0 give $$ W_a(K)-W_b(K)\ll_{A,L}{N\over\log^A x}. \tag{3} $$ Expand k=ℓr with ℓ>V and weight Λ(ℓ). First remove terms with (ℓ,Q)>1. Then ℓ=p^j for p|Q, and their total mass is at most $$ \sum_{p\mid Q}\sum_{p^j>V}{N\log p\over p^j} \ll {N\log Q\over V}, \tag{4} $$ which is negligible at every fixed logarithmic scale. For the remaining terms, (k,Q)=(r,Q)=g. For each such r the progression condition on ℓ is a primitive residue class modulo Q/g, namely (a/g)(r/g)⁻¹ modulo Q/g. Removing prime powers of primes dividing g changes the total by at most (4). The lengths of the ℓ intervals are independent of a within this gcd class. The Siegel–Walfisz error, summed over r≤N/V, is O_L(N log x exp(−c_L√log V)), hence gives (3). The theorem applies because every nonempty ℓ interval has endpoints at least V, a fixed positive power of x, while Q/g is at most a fixed power of log x. Increasing the fixed logarithmic exponent in its modulus condition absorbs the constant between log V and log x. Its constants need not be effective for this asymptotic specification. Imported source: [Tao, Notes 2, Exercise 64, Siegel–Walfisz](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/). The β convolution and nonprimitive-residue reduction are included above. By (2) and (3), uniformly in d and K, $$ \sum_{k\in K}\beta_V(k)r_Q(dk) \ll_{A,L}{N\over\log^A x}. \tag{5} $$ Indeed one can subtract the average W in each gcd class. The cost of summing Q residue classes and of the local factors is only a fixed logarithmic power: |r_Q|≤2Q/φ(Q)≤2Q. Absorb it by choosing a larger exponent in (3). ## 3. The local bilinear sum is controlled Define the same bilinear functional on the joint local model: $$ B_Q(x)=\sum_{d>U,\ k>V\atop x/20, $$ B_Q(x)\ll_{H,L}{x\over\log^H x}, \tag{7} $$ uniformly for all the specified Q. No cancellation of μ(d) is needed in this step. This is a theorem about the local model, obtained from a classical prime-progression theorem; it is not a theorem about B(x). ## 4. The comparison part of reconstruction is also controlled For an arithmetic function F, write 𝓑(F)=Σ_{d>U,k>V,dk∈J_x}μ(d)β_V(k)F(dk). In fact, for every H>0, $$ \mathcal B(b),\ \mathcal B(b_Q) \ll_H x/\log^H x, \tag{8} $$ uniformly in the choice of the finite prime set for b_Q. This part of the reconstruction need not be left open. Here is a proof with uniformity in k explicit. For any set R of odd primes, including the full set, let f_R(k)=∏_{p|k,p∈R}(p−1)/(p−2). For odd k, factor f_R(dk)=f_R(k)∏_{p|d,p∈R,p∤k}(p−1)/(p−2). The Dirichlet series of the d coefficient $$ g_{k,R}(d)=1_{d\text{ odd}}\mu(d) \prod_{p\mid d\atop p\in R,\ p\nmid k}{p-1\over p-2} $$ equals ζ(s)⁻¹H_{k,R}(s), where $$ H_{k,R}(s)=(1-2^{-s})^{-1} \prod_{p\in R\atop p\nmid k} \left(1-{p^{-s}\over(p-2)(1-p^{-s})}\right). $$ If h_{k,R}(e) are its coefficients, then Σ_e|h_{k,R}(e)|/√e is bounded by one constant, uniformly in k,R: the odd-prime absolute Euler factors are 1+O(p^(-3/2)), and the factor at 2 converges. The convolution g=μ*h and the classical bound Σ_{n≤t}μ(n)≪_A t/log^A t therefore give, uniformly, $$ \sum_{d\le T}g_{k,R}(d)\ll_A T/\log^A T. $$ Indeed split e at √T: on the smaller part apply the Möbius bound, and on the larger use Σ_{e>√T}|h(e)|/e≪T^(-1/4). For the d interval in 𝓑, all relevant endpoints are at least U or yield an empty interval. Thus its sum is ≪_A(x/k)/log^A x. Multiplying by β_V(k)≤log x and f_R(k)≤f(k)≪log log x, then summing k, loses only O(log²x log log x). The bounded constants 2C₂ and 2C_Q do not affect the argument. Even k give zero. This proves (8). The imported Möbius bound is the q=1 case of [Tao, Notes 2, Exercise 66](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/). It is applied only after the coefficient is separated multiplicatively; it cannot be applied in the same way to Λ(dk−2). ## 5. The reconstruction error cannot be hidden The exact missing term is $$ E_Q(x)=B(x)-B_Q(x) =\sum_{d>U,\ k>V\atop dk\in J_x}\mu(d)\beta_V(k) \left[(\Lambda(dk-2)-A_Q(dk))-(b(dk)-b_Q(dk))\right]. \tag{9} $$ In view of (7), the sufficient H_B condition is equivalent to E_Q≥−(1−η)C₂x+o(x) on the chosen scales. It is still **OPEN**. The local theorem has not estimated this signed reconstruction term. By (8), its comparison part is negligible. Equivalently, $$ E_Q=\mathcal B(a-A_Q)+O_H(x/\log^H x),\quad \mathcal B(A_Q)=O_{H,L}(x/\log^H x),\quad B=\mathcal B(a)+O_H(x/\log^H x). \tag{10} $$ Thus the unresolved arithmetic is specifically μ(d)β_V(k) against Λ(dk−2), or against its difference from the local roughness model. All the comparison and polylogarithmic local terms have been estimated separately. Moreover an absolute approximation cannot be o(x): uniformly in this Q class, $$ \sum_{n\in J_x}|w(n)-r_Q(n)|\ge(\tfrac12+o(1))x. \tag{11} $$ To prove (11), restrict the sum to n−2 prime. On these n, w(n)−r_Q(n)=log(n−2)+O_L(log log x). For A_Q and b_Q, this bound follows from Q/φ(Q)≪log Q≪_L log log x. Also $$ b(n)\le2{n\over\phi(n)}\ll\log\log(3n), $$ since f(n)/(n/φ(n))≤C₂⁻¹ on odd n. The standard totient bound follows by splitting prime divisors at log n and using Mertens for the smaller ones. PNT now gives Σ_{n∈J_x,n−2 prime}log(n−2)=x/2+o(x); the O(log log x) correction per prime totals o(x). This proves (11). Inequality (11) does **not** say that E_Q has size at least x. Its coefficients are signed, so weighted cancellation could still be substantial. It refutes the proposed o(x) unweighted absolute approximation, not a signed transfer. ## 6. What has been settled Joint local folds up to polylogarithmic combined modulus, their β-weighted balance, their local bilinear cancellation, and the comparison terms are established. The first absolute reconstruction strategy fails by (11). Any further use of these models must supply a signed bound for (9), retaining its factor and interval restrictions, or an independently justified alternative arithmetic input. More exact local recursions at these depths do not by themselves supply that bound. The true bilinear remainder and twin-prime infinitude remain open.