Prime-Power Activation Geometry in Finite-Scale Weil Operators: Quadratic Rank, Intrinsic Curvature, and an Arithmetic Weyl Law

We develop a finite-scale structural theory for the quadratic prime-power activation walls associated with Weil-type arithmetic operators. The arithmetic coefficient at a prime power $q = p^m$ is $c_q = \Lambda(q)/\sqrt{q}$, with activation location $L_q = \frac{1}{2}\log q$. Starting from the finite truncated-shift model itself, we derive the quadratic wall jet and the distinct-prime-base product-wall amplitude. The activation wall has rank at most two and, for every noninitial strict wall, rank exactly two in the minimal even block, whereas a pure product ghost is rank one. The canonical even contrast gives the exact arithmetic coordinate $A(q) = 48 c_q R_\Lambda(q)/(\log q)^3$, where $R_\Lambda(x) = \sum_{r < x} \Lambda(r) \log(r) \log(x/r)/\sqrt{r}$. On $\operatorname{span}\{P_0, P_2\}$, the standard $L^2(-1, 1)$ metric turns the wall bilinear form into the intrinsic scalar $\Delta_{\mathrm{geom}}(q) = -\det(G^{-1}B_q) = \frac{5}{4}A(q)^2$. The prime number theorem then yields $\sum_{p^m \le X} A(p^m)^2 \sim 36864 X/(\log X)^3$ and $\sum_{p^m \le X} \Delta_{\mathrm{geom}}(p^m) \sim 46080 X/(\log X)^3$, with higher prime powers contributing $O(\sqrt{X}/(\log X)^3)$. The paper is deliberately structural. It makes no claim toward the Riemann hypothesis, Montgomery pair correlation, a support-greater-than-one estimate, or an all-order strict-rank theorem. Higher-order boundary-history rank results belong to a separate companion line and are not used here.