# Factor signs in the complete global remainder **Twin-prime infinitude remains OPEN; this attempt gives no improved signed bound.** It derives a factor formula and pays its exceptional terms, then refutes a particular proposed upper bound on negative mass. The next estimate is stated with the full small-prime sum and the prime filters intact. This is not a novelty claim. Baseline: commit 689dac9, 2026-09-06. The parameters, uniform global identity and consumer are those of [global-cutoff-averaging.md](global-cutoff-averaging.md). In particular W_L=b_L=floor(x^(6/25)), W_R=b_R=floor(x^(1/20)), and F_i(m)=sum_(d|m) mu(d) rho_i(d). All assertions below concern sufficiently large x, so both inputs in J_x and J_x-2 exceed both W_i. ## 1. Exact factor formula, including the exceptional term Suppress the side index. Factor uniquely \[ n=s_W(n)t_W(n),\qquad s_W(n)=\prod_{p\le W}p^{v_p(n)},\qquad t_W(n)=\prod_{p>W}p^{v_p(n)}. \tag{1} \] Write s=s_W(n), t=t_W(n), and H_W(n)=log t. Empty products are one. Since rho(d)=0 for d>=b=W, no nonzero divisor term in F contains a prime exceeding W. Consequently F(n)=F(s), including repeated prime factors, and \[ F(s)=\sum_{A\subseteq\{p:p\mid s\}} (-1)^{|A|}\rho\left(\prod_{p\in A}p\right). \tag{2} \] This is the complete subset sum, not a truncation by the number of prime factors. F depends only on the radical of s. The convolution identity in the preceding note also reads \[ G(n)=\Lambda_{>W}(n)- \sum_{\substack{r\mid n\\r>W}}\Lambda(r)F(n/r). \tag{3} \] For r=p^j with p>W, removing any number of copies of p leaves F unchanged. The sum of their Lambda weights is log t. Thus exactly, \[ \boxed{G(n)=\Lambda_{>W}(n)-H_W(n)F(s)-E_W(n),} \qquad E_W(n)=\sum_{\substack{p\le W,\ j\ge2\\p^j\mid n,\ p^j>W}} (\log p) F(n/p^j). \tag{4} \] No prime powers inside beta were replaced by primes. The Lambda term is the r=n correction; in particular G(p)=0 at every prime. Call n regular for W when p^(v_p(n))<=W for every p<=W dividing n. Then E_W(n)=0. On a regular composite which is not a prime power, \[ G(n)=-(\log t)F(s). \tag{5} \] For a regular proper prime power n=p^k>W, necessarily p>W, and G(n)=-(k-1)log p. Therefore it too has the negative sign of -F(s), since s=1 and F(1)=1. These inputs must not be assigned -k log p without an error calculation. Thus, on regular composite inputs with t>1, a negative coefficient product occurs exactly when the two F values have opposite signs. The positive rough weights only change its size. Primes have zero G; smooth inputs with t=1 and regularity also have zero G. This classification includes all regular squareful composites. ## 2. The exceptional terms have a power saving on the full pair For each prime p<=W let k(p) be the least integer with p^k(p)>W. Then k(p)>=2 and every irregular n is divisible by some p^k(p). Split the reciprocal sum at sqrt(W). For p<=sqrt(W), each summand is less than 1/W and there are at most sqrt(W) choices. For larger p, k(p)=2, and the sum is bounded by the tail of sum_(m>sqrt(W)) m^-2. Hence, using only the union bound, \[ \#\{n\le x:n\text{ irregular for }W\} \le x\sum_{p\le W}p^{-k(p)}\ll xW^{-1/2}. \tag{6} \] There is no squarefree restriction on the remaining integers. Pointwise beta_W(m)<=log m, |F(m)|<=tau(m), and the definition of G gives |G(n)|<=tau(n)log n. Also |H_W(n)F(s)|<=tau(n)log n and |E_W(n)|<=tau(n)log n by summing the prime-power Lambda weights. The elementary divisor bound tau(n)<<_epsilon n^epsilon therefore pays the difference of full products on the union of the two exceptional sets: \[ O_\epsilon\left(x^{39/40+\epsilon}\right). \tag{7} \] Here the worse cutoff is W_R approximately x^(1/20); absorb both pointwise divisor factors and logarithms into the displayed epsilon. The number of proper prime powers up to x is O(sqrt(x) log x), by summing the integer bases for exponents at least two. Their full product correction is O_epsilon(x^(1/2+epsilon)), also within (7). The same reasoning applies to positive and negative parts, since z -> max(z,0) is Lipschitz with constant one. For clarity define the explicit all-input coefficient \[ D_i(n)=H_{W_i}(n)F_i(s_i(n)),\qquad C_i^{\rm comp}(n)=\mathbf1_{\{n\ {\rm composite}\}}D_i(n). \tag{8} \] This C^comp is not the sharp corner coefficient C in the previous notes. Equations (4)--(7) give the full statement \[ \mathcal R(x)= \sum_{n\in J_x} C_L^{\rm comp}(n)C_R^{\rm comp}(n-2) +O_\epsilon(x^{39/40+\epsilon}). \tag{9} \] Fix any epsilon<1/40 when consuming this error; it is then smaller than x/log^A x for every fixed A. No irregular, squareful, or prime-power term is silently discarded. ## 3. A proposed pair-trigger majorant fails A natural first attempt was to charge F(s)^-=max(-F(s),0) to pairs of small primes whose product crosses the initial cutoff. A concrete version would require some fixed K with \[ F(s)^-\ \le\ K\sum_{\substack{pa\}}. \tag{10} \] Even restricting the individual primes to p<=a does not repair this inequality. Take ten distinct small primes such that every triple product is below a and every quadruple product is above b. All the divisor profile values in (2) are then zero or one, and \[ F(s)=1-10+45-120=-84. \tag{11} \] Every pair is below a, so the right side of (10) is zero for every K. For n=sq with q>W prime and all ten small primes <=W, (4) gives G(n)=84 log q. This is a positive coefficient caused by higher-order subset cancellation, not by a pair crossing a. The configuration fits strict margins in the actual left exponent ranges: choose the ten prime exponents in (.069,.071). Triple products have exponent at most .213<.22, and quadruple products have exponent at least .276>.24. Their product has exponent in (.69,.71), leaving a large prime cofactor near exponent .30, above .24. The usual [prime number theorem, Tao Notes 2, (3)](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/) supplies distinct primes in these fixed power intervals and a prime q in (x/(2s),x/s] for sufficiently large x. Only this existence statement uses that standard input; no shifted occurrence or density is claimed. Floors preserve the strict margins eventually. An independent finite integer instance uses the ten primes 101,103,107,109,113,127,131,137,139,149, with a=4,000,000, b=W=6,000,000 and q=10,000,019. The validator checks primality, all subsets, (11), and the zero pair count exactly with integer arithmetic. These finite cutoffs are a proxy; the exponent argument above gives the separate asymptotic admissibility check. This refutes (10) and any bound that vanishes whenever all pairs are below a. It does not refute a higher-order majorant, a different signed inequality, or the use of small-prime structure. Nor does the single-input cell prove that it occurs together with a particular factorization of n-2. ## 4. A valid negative-part inequality and its missing estimate The [switching follow-up](switching-negative-mass.md) now proves that the negative-only target (15) below is false at the present cutoffs: an actual composite-filtered family already has negative mass with liminf at least kappa*C2*x, kappa>3/2. This holds for both the linear and C3 profiles. The inequalities and prime-filter algebra here remain valid; use the net consumer retaining P_comp. Let P_G and N_G be the positive and negative parts of the original full G product. Define D_i^+=max(D_i,0), D_i^-=max(-D_i,0), and \[ N_D(x)=\sum_{n\in J_x} \big[D_L^-(n)D_R^+(n-2)+D_L^+(n)D_R^-(n-2)\big]. \tag{12} \] Removing the prime inputs in (8) gives the exact correction \[ \begin{split} A_{\rm pr}(x)={}& \sum_{\substack{n\in J_x\\n\ {\rm prime}}} (\log n)D_R^-(n-2)\\ &+\sum_{\substack{n\in J_x\\n-2\ {\rm prime}}} (\log(n-2))D_L^-(n). \end{split} \tag{13} \] Both primes exceed the cutoffs, so D_i(p)=log p. A double-prime input contributes zero to (12) and to (13); there is no missing intersection correction. Therefore \[ N_G=N_D-A_{\rm pr}+O_\epsilon(x^{39/40+\epsilon}),\qquad N_G\le N_D+O_\epsilon(x^{39/40+\epsilon}). \tag{14} \] This is a valid upper bound, not the desired bound. To use the negative part alone, one would still need, for fixed c,K>0 on unbounded common dyadics, for example \[ N_D-A_{\rm pr}\le C_2x-cx/\log^K x. \tag{15} \] More generally retain the corresponding composite positive sum P_comp from (9) and prove N_D-A_pr-P_comp <= C2*x-c*x/log^K x. Its error is already paid. Neither inequality is established here. The available linear Type I estimates do not evaluate (12) or (13): taking a positive or negative part of a divisor sum is nonlinear. Replacing F^- by a short signed divisor polynomial requires a pointwise bound with a stated remainder and support. No such replacement has been supplied. The support of the original linear F cannot simply be assigned to its negative part. ## 5. Full cofactor form and the next bounded question The subsequent [smooth-majorant argument](global-smooth-majorant.md) completes a higher-order bound: a different C3 admissible profile has full absolute O(x) mass. The cofactor formula below remains valid for that profile, with hats; its signed constant is OPEN. For any s,t let F_L(s),F_R(t) keep their complete subset sums. The main sum in (9) is exactly \[ \sum_{\substack{su-tv=2,\ su\in J_x\\ P^+(s)\le W_LW_i is strictly greater than x^(w_i), and the products are at most x. This does not bound the number of prime factors in s or t. The remaining signed question is whether a signed Buchstab decomposition of the rough factors in (16), or a justified majorant/minorant of the full F sums, can price the mixed-sign mass together with (13) and retained positive mass at scale x. Keep the small-cofactor sum inside until the new inequality is justified. A calculation on only two or three small factors is not a full estimate; it needs a bound for everything omitted. The subsequent absolute O(x) bound, without a sufficient constant or signed improvement, does not meet the consumer. Changing admissible averaging profiles changes this sign decomposition, but their full signed residuals all equal S-C2*x+O_A(x/log^A x) by the uniform global reduction. Thus choosing a profile can facilitate an estimate; it cannot be asserted to improve the underlying signed total. ## 6. Falsification and reproducibility [global-factor-signs-validation.js](global-factor-signs-validation.js) checks the actual logarithmic ramp using exact integer polynomials in formal prime logarithms, after multiplying by log(b/a). It compares the original divisor convolution against (4), checks the prime zeros and large-prime-power correction, and tests (9) exactly off its explicitly identified exceptional sets. Squareful regular inputs are retained. It also exercises active failures when E or the prime correction is omitted. Positive/negative splits in the small scan use floating evaluation, separately labeled; the ten-prime counterexample uses exact integers. The new algebra would fail on a mismatch in these identities; those finite tests passed. The uniform error claim is supported by (6) and the pointwise divisor bounds, not by fitting finite counts. The pair-trigger inequality has an explicit counterexample. No asymptotic sign density, estimate for (15), or sufficient twin margin is tested or claimed by this script.