# Literature structures for the signed endpoint remainder **Checked 2026-09-06. No twin-prime lower bound is claimed.** The [grouped-divisor follow-up](grouped-divisor-moment.md) consumes this audit's proposed moment and derives an additional controlled region. This note is a theorem-interface audit and a derivation of reusable kernel identities. It is not an exhaustive bibliography, a proof of optimality, or a claim that no other theorem applies. The relevant successes and unsuccessful imports share one entry in [OUTCOMES](OUTCOMES.md). ## 1. What an imported result must actually do [Residual coverage](residual-coverage.md), (16)--(20), is the current specification. With w=6/25 and v=1/20 the original divisor exponents lie in delta in [0.24,0.76] and nu in [0.05,0.95]. The controlled region is delta+nu<0.76 or 5delta+2nu<2.46. The concrete remaining E_* retains de>floor(x^0.75) and d^5*e^2>floor(x^2.45), including the margin strips. For every fixed H the named inputs give \[ \sum_{x/2=c*x/log^K x on unbounded dyadic scales, for fixed positive c,K. The precise rescaled average in [endpoint-target-audit.md](endpoint-target-audit.md) is also sufficient. An estimate on unweighted Liouville correlations is a different premise. At delta=nu=2/5 the expanded lengths are M=x^(16/25), N=x^(9/20). The critical harmonic length is A~MN/x=x^(9/100). The old right block budgets are 0.895 (nonzero completion), 1.02 (equal frequency after an extra Cauchy inequality), and 0.84 (complete periods). Thus improving only the first term cannot close this box. In the specified right first-power sector the signed cross-divisor moment needs an upper bound O(x^(34/25-epsilon)); the current bound has exponent 7/5. More than 1/25 in the moment exponent is required. A moment improvement contributes half as much to the final block exponent. Every proposed import must retain composite moduli, actual coefficient norms, both gcd branches, endpoint weights and their variation, complete periods, and uniformity in the divisor twists used for exact cuts. The full positive majorant already has a separate bound in [endpoint-pairing.md §5](endpoint-pairing.md). ## 2. The exact dictionary: common factors, frequencies and signs This section is derived here from the native phase, independently of the newer theorem inputs. Write e_c(t)=exp(2*pi*i*t/c), theta=2/g, g in {1,2}. For two reduced right divisors u1=e1*q1 and u2=e2*q2 put \[ j=(u_1,u_2),\qquad u_i=j\ell_i,\qquad(\ell_1,\ell_2)=1, \qquad c=[u_1,u_2]=j\ell_1\ell_2. \] For (m,c)=1, the two inverse phases combine exactly as \[ e_{u_1}(-\theta h_1\bar m) \overline{e_{u_2}(-\theta h_2\bar m)} =e_c(r\bar m),\qquad r=-\theta(h_1\ell_2-h_2\ell_1). \tag{1} \] Indeed the inverse modulo c reduces to each smaller modulus, and c/u1=ell2, c/u2=ell1. The endpoint factor multiplying (1) is \[ \Phi_{u_1,h_1}(m)\overline{\Phi_{u_2,h_2}(m)},\quad \Phi_{u,h}(m)=e\left(\frac{hz_0}{gmu}\right) -e\left(\frac{hz}{gmu}\right). \tag{2} \] The reciprocal orientation instead uses the opposite inverse sign and both shifted endpoints z0-2,z-2. Formula (1) also applies to general aggregated divisors u, without a squarefree assumption. **Equal frequencies.** For positive harmonics, \[ r=0\ \Longleftrightarrow\ (h_1,h_2)=t(\ell_1,\ell_2),\qquad t\in\mathbb Z_{>0}. \tag{3} \] This follows from coprimality of the ell_i. If u_i~N and h_i~A, such pairs require j>N/(2A). At the benchmark this means j of scale at least x^0.36, up to constant factors. Equal frequencies can have different original divisors and different primes: u1=6,u2=10, h1=3,h2=5 is an example. Zero *integer numerator* in (3) must be distinguished from a nonzero numerator divisible by c; the latter still needs its full gcd loss. A useful elementary count is \[ \#\{u_i\in(N,2N],h_i\in(A,2A]:h_1u_2=h_2u_1\} \le 8NA\,H_{\lfloor2\min(N,A)\rfloor}, \tag{4} \] where H_k=sum_(s<=k)1/s, for N,A>=1. To prove this, write the solutions as u_i=j*ell_i,h_i=t*ell_i with coprime ell_i. For k=max(ell1,ell2) there are at most 2k ordered pairs; the numbers of possible j,t are at most floor(2N/k),floor(2A/k). Sum 8NA/k. Dropping the lower interval endpoints only enlarges the count. This is a derived count, not an estimate for the entire signed moment. **Common-factor signs.** In the squarefree first-power sector, q_i is odd, q_i does not divide e_i, and mu(g*e_i) is nonzero. Consequently u_i is squarefree and j,ell1,ell2 are pairwise coprime. Then \[ \mu(ge_1)\mu(ge_2)=\mu(u_1)\mu(u_2) =\mu(\ell_1)\mu(\ell_2). \tag{5} \] For g=2 the live e_i are odd; the two mu(g) factors cancel. Thus there is no oscillating mu(j) left in this particular product. There are still squarefreeness restrictions on j, prime-divisor conditions, interval cuts, logarithmic weights and divisor twists. Formula (5) does **not** make the full modulus weight constant or well-factorable. **Nonzero gcds.** With R=h1*ell2-h2*ell1 nonzero, \[ (\theta R,j\ell_1\ell_2) \le2(R,j)(h_1,\ell_1)(h_2,\ell_2). \tag{6} \] Use (R,ab)<=(R,a)(R,b), then coprimality of ell1,ell2. This inequality does not require j coprime to either ell_i, so repeated prime powers are included. For J>=1 and 0<=sigma<=1, \[ \sum_{J=constant*L². Its exceptional-spectrum bound has right side (qaH)^epsilon*(1+aK/q)*(||a_n||²+gcd(a,q)*K*(H/L+H²/L²)), where n~K. The permitted spectral weight is \[ X_{\rm sp}\ll\max(1,q/(aK)) \max\left(1,\frac{KH}{(H+L)L\mathcal D}\right),\quad \mathcal D=\min_{t\ge1,\ i=1,2}(t+(K/L)\|t\alpha_i\|). \] Theorem 2 handles single exponential sequences. These are estimates for the exceptional Maass spectrum; a full Kuznetsov application also has regular, continuous and, where present, holomorphic contributions. **Our translation and a limitation.** Equation (1) supplies the same linear difference after swapping the ell labels. On j~J, take L=N/J, H=A, numerator length K~HL. A possible level is q~L², but its actual Kuznetsov realization still needs proof. Even granting that realization, a=1 and the most favorable phases alpha_i=0, when H<=L the additional factor is comparable to max(1,H²/L). For L>=H² it contributes no power improvement over the general large sieve's max(1,q/K,q²/K³). At the benchmark L=x^0.45/J,H=x^0.09. Thus **J<=x^0.27 has no such structured advantage** in this diagnostic. Smaller numerator bands do not improve that factor. Large J may benefit, but its contribution is also cheaper in the provisional classical-completion budget below. Arbitrary spectral levels, new factorizations or another orientation could change this conclusion. It is not a closure of spectral methods. Before applying this source, construct the modulus average, price the squarefree j weight and all endpoint separations, then compare total budgets. An earlier G₂ theorem comparison did not perform this match. ### B. Composite-modulus bilinear sums and quadratic characters [Blomer--Pascadi, *Bilinear forms with Kloosterman sums via quadratic characters*, arXiv:2607.24311v1](https://arxiv.org/html/2607.24311v1), 27 July 2026 preprint, Theorem 1.1, applies to all positive moduli c, separate complex coefficient sequences on intervals of length <=B<=c, and a unit a modulo c. With (m,n,c)=1 its bound for sum alpha_m beta_n S(am,n;c) is \[ \|\alpha\|_2\|\beta\|_2c^{1+o(1)} (B^{1/8}c^{-3/32}+B^{5/16}c^{-3/16}+B^{2/3}c^{-7/18}). \] At B=sqrt(c) this saves c^(-1/32). The coprimality constraint can be omitted for both intervals {1,...,B}. Theorem 5.5 gives an unequal-length variant. Theorem 1.6 also improves a general exceptional-spectrum bound. **Match still missing:** (8) has coupled Fhat and variable c; neither is covered just by choosing separate alpha,beta. At small J, c~x^0.9/J. Even a hypothetical one-for-one saving c^(-1/32) at J~1 corresponds to only x^(-9/320), less than the old required moment saving x^(-1/25). This is a budget warning, not a legal application or a bound on what repeated amplification could accomplish. Investigate after locating a genuinely separated short bilinear piece. [Pascadi, *Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums*, published version](https://link.springer.com/article/10.1007/s00039-026-00746-0), Theorems 1.1--1.2, offers another interface: general-modulus savings and stronger savings using a suitable factorization of c. For squarefree c with a balanced divisor, Theorem 1.2's displayed factor can save c^(-1/12) at square-root lengths. This is potentially more useful than the uniform exponent, but the required lengths, coprimality and coefficient separation must be paid. Its Lemmas 3.2--3.3 are the classical Ramanujan/Weil inputs already used here. The full-dual-spectrum operator obstruction in [prime-band-completion.md](prime-band-completion.md) remains valid at its stated scope; it does not exclude a restricted structured application. ### C. Centered dispersion and convolution structure [Pascadi, *Smooth numbers in arithmetic progressions to large moduli*, arXiv text](https://arxiv.org/html/2304.11696), Theorem 4.1 and §5, treats convolutions of three bounded sequences with explicit restrictions on their lengths and the modulus range. Its power-saving error subtracts **all characters of conductor <=D**, not just the principal character. For D=1 the displayed error does not give a power saving. The deamplification step retains the centered combination S1-2*Re(S2)+S3 before estimating it. The journal version numbers the triple-convolution theorem differently; use the arXiv numbering here. This suggests checking whether the density comparator can be retained inside an earlier dispersion moment, or whether a useful extra factor can be exposed. Neither our determinant equation nor (5) automatically provides the required three independent sequences. Small-conductor character terms would become an additional obligation, not a free main term. This is a structural alternative if direct completion stalls. [Fouvry--Radziwiłł, *Level of distribution of unbalanced convolutions*, Theorem 1.1](https://arxiv.org/pdf/1811.08672), is a second interface: it keeps an explicit progression-distribution error E*(beta,N,Q) and requires an unbalanced length range, including M>Q*(MN)^epsilon. That error must be bounded for the actual coefficient. Its improved distribution level is not a theorem for every bounded convolution. [Pascadi, *On exponents of distribution of primes and smooth numbers*, Definition 1.1 and Theorem 1.3](https://arxiv.org/html/2505.00653), gives fixed-residue prime distribution with triply well-factorable weights through X^(5/8-epsilon). The definition requires bounded convolution representations for **every** factorization of the level into three prescribed sizes. One convenient factorization does not suffice. The associated twin-prime application is an upper bound. Neither a missing lower bound nor the needed well-factorability of our signed weights is supplied. This extends the line of [Maynard's well-factorable estimates](https://arxiv.org/abs/2006.07088) (abstract used for this bibliographic link only). ### D. Trace functions and multiplicative orthogonality [Fouvry--Kowalski--Michel--Sawin, *Bilinear forms with trace functions*, arXiv:2511.09459v3, Theorem 1.1](https://arxiv.org/html/2511.09459v3), gives power savings for prime-modulus trace-function bilinear sums with M,N<=q/2, M>=q^delta and MN>=q^(3/4+delta). Its simplified statement requires irreducible geometric monodromy, with a simple algebraic identity component or a finite quasisimple group. The bounded-complexity sheaf and those hypotheses must be identified for the actual transformed kernel. Our variable composite lcm modulus is not that input. CRT alone does not control the resulting weights or establish the geometric hypothesis. Retain this as a possible tool for a prime-modulus piece, not for (8) without further work. [Korolev--Shparlinski, *Sums of algebraic trace functions twisted by arithmetic functions*, Theorem 2.1](https://arxiv.org/pdf/1804.01337), does retain mu(n): for a nonexceptional isotypic trace function of bounded conductor modulo prime p, and N>=p^(1/2+epsilon), it bounds the sum by O(epsilon^(-1)*N*loglog(p)/log(p)). This is a **logarithmic** saving. It cannot by itself pay the fixed exponent deficit of the benchmark, even if the modulus and function hypotheses were supplied. No statement here excludes a later use at a logarithmic boundary. [Bourgain--Sarnak--Ziegler, *Disjointness of Möbius from horocycle flows*, Theorem 2](https://arxiv.org/pdf/1110.0992), replaces multiplicative orthogonality by estimates of F(p1*m)*conj(F(p2*m)) for distinct small primes. It gives a concrete alternative way to preserve mu instead of discarding it by Cauchy. The paper's horocycle consequence concerns a fixed dynamical system; it gives no uniform rate for our growing moduli and scale-dependent observables. A determinant-2 matrix description is not by itself a transfer to that theorem. Use the criterion only after exhibiting and bounding its dilation correlations. ### E. Divisibility graphs, parity correlations and weaker averages [Helfgott--Radziwiłł, *Expansion, divisibility and parity*, Main Theorem](https://arxiv.org/html/2103.06853), studies the centered operator \[ Af(n)=\sum_{p,\sigma=\pm1} (1_{p\mid n}-1/p)f(n+\sigma p) \] on an interval, with endpoint restrictions. After deleting a quantified exceptional set its norm is O(sqrt(K*sum_p 1/p)), for the stated subpower-sized prime ranges. This is a concrete link between exact divisibility arrangements and signed correlations. It uses centered adjacency and trace estimates, not independent random prime strikes. Our endpoint operator has different edges and weights. Applying this method would require a transfer identity, a norm estimate and a bound for the prime-detection mass on the deleted set. [Pilatte, *Improved bounds for the two-point logarithmic Chowla conjecture*, Theorem 1.1](https://arxiv.org/html/2310.19357v2), uses this direction of ideas, including products of primes and non-backtracking methods, to obtain sum_(n<=x) lambda(n)*lambda(n+1)/n=O((log x)^(1-c)) for some c>0. The prime-conditioned quantity we need is not that correlation. This is the most relevant longer-term connection to the original fold viewpoint, but it currently has a larger translation gap than (1)--(8). [Tao--Teräväinen, *Quantitative correlations and some problems on prime factors of consecutive integers*, arXiv:2512.01739v2, Theorem 3.1](https://arxiv.org/html/2512.01739v2), April 2026 version, gives quantitative correlations of 1-bounded multiplicative functions outside a small set of logarithmic scales. Its nonpretentious/equidistribution axioms are explicit; progression moduli and shifts are bounded by a small power of a parameter <=log X. Remark 3.2 includes Liouville correlations for linear forms with polylogarithmic coefficients. This quantifier pattern is relevant to our averaged sufficient target. Our polynomially growing coefficients, inverse-residue phases and truncated divisor weights do not satisfy that interface merely by normalization. An average over scales does not remove these discrepancies or replace prime detection by parity. ### F. Asymptotic sieve: the consumer, not a missing input for free [Friedlander--Iwaniec, *Asymptotic sieve for primes*, Theorem 1 and hypotheses (R),(B)](https://arxiv.org/pdf/math/9811186), turns distribution plus a signed bilinear hypothesis into a prime asymptotic for a nonnegative sequence. The displayed distribution level satisfies x^(2/3)