The Fej\'er-Dirichlet Lift: Entire Functions and $\zeta$-Factorization Identities

A Fej\'er-Dirichlet lift is developed that turns divisor information at the integers into entire interpolants with explicit Dirichlet-series factorizations. For absolutely summable weights the lift interpolates $(a*1)(n)$ at each integer $n$ and has Dirichlet series $\zeta(s)A(s)$ on $\Re s>1$. Two applications are emphasized. First, for $q>1$ an entire function $\mathfrak F(\cdot,q)$ is constructed that vanishes at primes and is positive at composite integers; a tangent-matched variant $\mathfrak F^{\sharp}$ is shown to admit an explicit, effective threshold $P_0(q)$ such that for every odd prime $p\ge P_0(q)$ the interval $(p-1,p)$ is free of real zeros and $x=p$ is a boundary zero of multiplicity two. Second, a renormalized lift for $a=\mu*\Lambda$ produces an entire interpolant of $\Lambda(n)$ and provides a constructive viewpoint on the appearance of $\zeta'(s)/\zeta(s)$ through the FD-lift spectrum. A Polylog-Zeta factorization for the geometric-weight case links $\zeta(s)$ with $\operatorname{Li}_s(1/q)$. All prime/composite statements concern integer arguments. Scripts reproducing figures and numerical checks are provided in a public repository with an archival snapshot.