Geometry and Constants in Finite Ring Continuum

We study the Euclidean prime-shell stage of the Finite Ring Continuum programme through a framed finite field and establish three main components. First, after fixing an extended frame, the shell supports a finite orbital complex combinatorially equivalent to the two-sphere; the labeled complex is frame-presented, while its cellular isomorphism class is shell-invariant. Second, its core Euclidean symmetry axes appear with increasing framing depth: additively framed negation, affinely framed reciprocal inversion, and Euclidean conjugation in the extended frame. Third, standard density and Lipschitz estimates show that sufficiently large prime-shell grids reproduce the observer-side outputs of any fixed bounded Euclidean experiment to any prescribed tolerance. The Fourier formalism is recorded strictly as a discrete Fourier transform over the shell ring, while contact with conventional continuum Fourier language is treated only as an observer-side large-prime approximation.