Toward a Native Multiplicative-Spectral Theory I: Admissible Eulerization Hosts for the Prime-Successor Anchor P and a C*-Spectral Ambient

This paper formulates the first admissibility layer on the $1 \leftrightarrow s$ edge after the two anchors have already been fixed. The multiplicative anchor is the prime-successor framework $P$ : a 1 -anchored atomic system generated by commuting prime-successor maps. The spectral anchor is a unital $C^*$-ambient, in which spectrum, invertibility, and functional calculus are stable and meaningful. The purpose is not to introduce a new universal object. The purpose is to specify the exact ambient requirements under which Euler-type multiplicative assembly can be realized, completed, extended, and tested for spectral obstruction once the multiplicative and spectral anchors are already chosen. The paper distinguishes four regimes. Finite prime-successor data with finite degree are formal and complete without genuine extension. The first real wall appears already for one prime-successor direction with infinite degree. This forces a local completion problem. $\Lambda$ finite family of such directions introduces no essentially new obstruction. The second real wall is the infinite-prime regime, where the global assembled object must be extended beyond literal convergence inside the chosen $C^*$-ambient. The outcome is an axiomatic notion of an admissible Eulerization host for the anchored pair ( $P, C^*$ ). This notion is intentionally framework-level: any realization satisfying the stated axioms is suitable for Eulerization in the present sense.