We develop a unified treatment of the Riemann zeros, of primitive Dirichlet -functions, and of Bj\"orner's complex of squarefree integers, based on the Klein group generated by and , and on the symmetric coordinate . Under the Cayley map , the critical line becomes the unit circle, and every cross-ratio of the orbit depends on the single invariant , where and . We introduce the -moments , with , and the gap hierarchy with and . We prove unconditionally that for every , with equality for each fixed if and only if the Riemann Hypothesis holds; each summand is locally proportional to the square of the Bernstein index. If RH fails, the rate is , where and is the smallest positive ordinate of an off-line zero. We give a non-asymptotic two-sided bound, valid at every finite , that makes the hierarchy a falsifiable numerical test. We extend the formalism to symmetric completions of primitive Dirichlet -functions, where the gap hierarchy is proved under the explicit hypothesis (H3') that every nontrivial zero satisfies . We prove an exact Chebyshev transform between the -moments and the Li coefficients, and an unconditional closed form for every -moment: with universal rational coefficients and , where are the generalized Euler--Kronecker constants. The case identifies with the Euler--Kronecker constant. In the third direction, we study Bj\"orner's complex of squarefree integers, prove that the Mertens invariant equals the index of the bipartite block of the prime-shift operator, obtain an exact kernel decomposition of the Mertens residual with critical exponent , and establish two independent structural obstructions to the spectral modification conjectured in earlier revisions of that work. Extended numerical verifications are reported. No proof of the Riemann Hypothesis is claimed.