Prime-Built Functions and Integer Hulls: Zeros, Counting Laws, Capture Mechanisms, and the Parity of the Prime Count

This paper unifies eight studies on functions built from the primes and on exact lattice-point counts for the primes. The analytic part is organised around the exponential atom $u_n(z)=\log n\cdot n^{-z}$. Its sum over integers gives $F(z)=\sum_{n\ge2}(n^{1/n^z}-1)$, its sum over primes gives $F_{\mathbb P}(z)=\sum_p(p^{p^{-z}}-1)$, and its product gives $G(z)=\prod_p\frac12(1+p^{p^{-z}})$. All three are nonlinear deformations of $\zeta'$-type objects. Three mechanisms recur: The Bohr torus. Every sharp abscissa computed is attained at one extremal character. In particular, we prove that the supremum of the real parts of the zeros of $\zeta'$ is $\sigma_1=2.81301402025\ldots$. Winding on the critical line. It yields an exact crossing identity for $F$ that reformulates the Riemann Hypothesis. It also shows that the zeros of $F_{\mathbb P}$ follow those of $\zeta'$ (under RH), with an integer excess equal to 12 up to height 1000. Local models. For $f=A+\chi A(1-z)$, which has the functional equation of $\zeta$ but no Euler product, a two-parameter capture law decides which zeros leave the critical line. With no fitted parameters it classifies 146 of 147 cases, and it extends to unrelated series and to the Davenport–Heilbronn function. Two companions isolate what depends on arithmetic. The first is a Dirichlet series with harmonic exponents, with a Papkovich–Fadle chain of zeros and Euler’s constant in its zero count. The second is the power series $\sum z^p$, whose truncations have a conjectured Mahler measure $\sim e^{-\gamma/2}\sqrt{\pi(N)/2}$. The algorithmic part computes $\pi(x)\bmod 2$ deterministically in $O(x^{5/12}\log^2 x)$ operations through a Möbius-free congruence. It also builds an exact three-dimensional integer-hull walk. Every statement carries a status label (proved, conditional, numerical or conjecture), and open problems are listed with their consequences.