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We develop a unified treatment of the Riemann zeros, of primitive Dirichlet $L$-functions, and of Bj\"orner's complex of squarefree integers, based on the Klein group generated by $s \mapsto 1-s$ and $s \mapsto \bar{s}$, and on the symmetric coordinate $\tau = s(1-s)$. Under the Cayley map $z = 1 - 1/s$, the critical line becomes the unit circle, and every cross-ratio of the orbit depends on the single invariant $\mathrm{CR}_{\mathrm{norm}} = \delta^2/(\delta^2 + \gamma^2)$, where $\rho = \sigma + \mathrm{i}\gamma$ and $\delta = \sigma - 1/2$. We introduce the $\tau$-moments $Q_n := \sum_{\gamma > 0} \mathrm{Re}\,\tau_\rho^{-n}$, with $\tau_\rho = \rho(1-\rho)$, and the gap hierarchy $\mathcal{G}_n := S_n - Q_n$ with $S_n := \sum A_\rho^{-n}$ and $A_\rho = 1/4 + \gamma^2$. We prove unconditionally that $\mathcal{G}_n \geq 0$ for every $n \geq 1$, with equality for each fixed $n$ 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 $\mathcal{G}_n = M A_0^{-n}(1+o(1))$, where $A_0 = 1/4 + \gamma_0^2$ and $\gamma_0$ is the smallest positive ordinate of an off-line zero. We give a non-asymptotic two-sided bound, valid at every finite $n$, that makes the hierarchy a falsifiable numerical test. We extend the formalism to symmetric completions $\Xi_\chi = \Lambda(s,\chi)\Lambda(s,\bar\chi)$ of primitive Dirichlet $L$-functions, where the gap hierarchy is proved under the explicit hypothesis (H3') that every nontrivial zero satisfies $|\mathrm{Im}\,\rho| \geq 1/2$. We prove an exact Chebyshev transform between the $\tau$-moments and the Li coefficients, and an unconditional closed form for every $\tau$-moment: $Q_n(\chi) = \sum_{k=1}^n c_{n,k} g_k$ with universal rational coefficients $c_{n,k} = (-1)^{k+1} \binom{2n-k-1}{n-k}/(k-1)!$ and $g_k = \delta_{k,1}\log(q/\pi) + 2^{1-k}\mathrm{Re}\,\psi^{(k-1)}((1+a)/2) - 2\mathrm{Re}\,\gamma_\chi^{(k)}$, where $\gamma_\chi^{(k)} = -(\log L)^{(k)}(1,\chi)$ are the generalized Euler--Kronecker constants. The case $n=1$ identifies $Q_1(\chi)$ with the Euler--Kronecker constant. In the third direction, we study Bj\"orner's complex $\Delta_n$ 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 $\alpha = 1/2$, 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.