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1 \pageheading{PRIME SIEVES\; /\; MULTIPLE PARAMETERS}{Overlap is part of the arithmetic}
2 For a finite set $F$ of distinct primes, define
3 \[
4 \nu_{F,\boldsymbol\lambda}
5 =\sum_{n\ge1}\mu(n)^2n^{-4}
6 \prod_{p\in F}(1+\lambda_p\mathbf1_{p\mid n})\delta_{x_n},
7 \quad x_n=(1-n^{-2})^2.
8 \]
9 For real strengths, the exact positivity domain is
10 \[
11 \boxed{\lambda_p\ge-1\quad\text{for every }p\in F.}
12 \]
13 Sufficiency is termwise positivity. If one strength is below $-1$, the
14 atom $n=p$ isolates its negative factor. The other strengths do not occur
15 at that atom. At every simultaneous deletion endpoint, primes outside
16 $F$ still supply infinitely many positive nodes, so all Hankel matrices
17 remain strictly positive definite.
18 
19 The exact coordinate formula is
20 \[
21 M_j(\boldsymbol\lambda)=\sum_{r=0}^{2j}(-1)^r\binom{2j}{r}
22 \frac{\zeta(4+2r)}{\zeta(8+4r)}
23 \prod_{p\in F}\left(1+\frac{\lambda_p}{p^{4+2r}+1}\right).
24 \]
25 Coefficients of the monic families lie in
26 $\mathbb Q(\pi^{-2},\lambda_p:p\in F)$.
27 For two primes, the overlap term is visible:
28 \[
29 M_j=M_j(0,0)+\lambda_pU_{p,j}+\lambda_qU_{q,j}
30 +\lambda_p\lambda_qU_{pq,j}.
31 \]
32 Here $U_{pq,j}>0$ is the $pq$-divisible submeasure moment. The family
33 does not reduce to adding separate prime responses without their overlap.
34 
35 \pageheading{PRIME SIEVES\; /\; FINITE TESTS}{Moment positivity is a weaker requirement}
36 \fig{12_multi_prime_domain}{The exact positive-measure domain and the
37 order-one moment test have different parameter geometry. Regions passing
38 a finite test outside the safe orthant still contain negative atoms.}
39 For primes two and three,
40 \[
41 M_0=105\pi^{-4}(1+\lambda_2/17)(1+\lambda_3/82).
42 \]
43 Both $(0,0)$ and $(-34,-164)$ have positive total mass. Their midpoint
44 $(-17,-82)$ has zero mass. Thus the parameter set passing the order-one
45 positive-definiteness test is nonconvex. This particular midpoint argument
46 does not establish nonconvexity of the semidefinite test, which permits zero.
47 For that test, take the two points $(0,0)$ and $(-34,-246)$.
48 The normalized masses $M_0/(105\pi^{-4})$ there are $1$ and $2$.
49 Their convex combination with weights $3/5$ and $2/5$ is
50 $(-68/5,-492/5)$, where the normalized mass is $-1/25$.
51 Thus the order-one semidefinite feasible set is also nonconvex,
52 by a separate counterexample.
53 
54 \claim{A separate additive family has different geometry}
55 For weights $1+\sum_{p\in F}\gamma_p\mathbf1_{p\mid n}$, positivity is
56 equivalent to
57 \[
58 1+\sum_{\gamma_p<0}\gamma_p\ge0.
59 \]
60 Every prime subset appears at its squarefree product, proving necessity.
61 Checking each atom proves sufficiency. Its finite Hankel matrices are affine
62 in $\gamma$, so the semidefinite feasible sets are convex spectrahedra.
63 The multiplicative family has genuine overlap terms and polynomial parameter
64 dependence. These are two distinct deformations, with distinct domains.
65 
66 \pageheading{PRIME SIEVES\; /\; CONSTRUCTIVE WITNESS}{Every negative prime atom has a finite detector}
67 For an integer $a\ge2$, let
68 \[
69 L_a(x)=\prod_{n=1}^{a-1}\frac{x-x_n}{x_a-x_n},\quad
70 C_a=L_a(1),\quad r_a=\frac{1-x_{a+1}}{1-x_a}<1,
71 \]
72 \[
73 f_{a,m}(x)=L_a(x)\left(\frac{1-x}{1-x_a}\right)^m.
74 \]
75 This polynomial kills every lower node, equals one at $x_a$, and has
76 absolute value at most $C_ar_a^m$ at every higher node.
77 \fig{13_signed_detector}{A rational detector for $p=3$, $\lambda_3=-1.1$,
78 $\lambda_2=1$. The first exact bound proving negativity is $m=5$, so a
79 Hankel matrix of order eight detects the signed defect.}
80 Set $C_F=\prod_{q\in F}\max(1,|1+\lambda_q|)$. The summable tail gives
81 \[
82 \int f_{p,m}^2\dd\nu
83 \le\frac{1+\lambda_p}{p^4}+
84 \frac{C_FC_p^2}{3p^3}r_p^{2m}.
85 \]
86 Hence the quadratic form is negative whenever
87 $pC_FC_p^2r_p^{2m}/3<-(1+\lambda_p)$.
88 For rational strengths this is an exact rational certificate. It needs no
89 numerical eigenvalue calculation. A depth tilt $x^k$ replaces the negative
90 atom contribution by $(1+\lambda_p)x_p^k/p^4$.
91 
92 \pageheading{PRIME SIEVES\; /\; FIBERS}{Real singularities survive multiple deletions}
93 Fix all strengths except $\lambda_p$, with the others at least $-1$.
94 The remaining moment matrix has the form $A_m+(1+\lambda_p)B_m$, where
95 $A_m$ is the $p$-free part and $B_m$ the $p$-divisible part. Both are
96 positive definite, even at other deletion endpoints: primes outside $F$
97 supply the first support, and integers $p\ell$ the second.
98 
99 Thus every one-variable determinant fiber has exact degree $m$ and all
100 roots real below $-1$. Its nearest margin satisfies
101 \[
102 \delta_m=\min_{\substack{f\in\mathbb R[x]\setminus\{0\}\\\deg f<m}}
103 \frac{\int f^2\dd\nu_{p\text{-free}}}
104 {\int f^2\dd\nu_{p\text{-divisible}}},\qquad\delta_m\downarrow0.
105 \]
106 The detector yields, for $C_\kappa=\prod_{q\ne p}\max(1,1+\lambda_q)$,
107 \[
108 \delta_{p+m}\le\frac{pC_\kappa C_p^2}{3}r_p^{2m}.
109 \]
110 The denominator retains the atom $p$; all lower nodes in the numerator are
111 killed. This extends the prime-two bound to every prime and to simultaneous
112 positive or deletion strengths. It does not assert simple roots or global
113 multivariate real stability.
114 
115 If at least two positive-domain strengths are nonzero, the complete monic
116 sequence cannot equal a one-prime sequence on the same variable and nodes.
117 Compact moment determinacy equates normalized measures; the fixed mass at
118 zero fixes their scale; atom weights at the individual primes then read off
119 every strength. An arbitrary affine change of variable is outside this separator.

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