Virtual Composite Modes and Arithmetic Product Walls in Finite-Scale Weil Gram Matrices

We study the finite arithmetic matrix generated by the prime-power shift operators in a compactly supported Weil quadratic form, focusing on structural changes as the support window crosses arithmetic thresholds. An exact four-strip decomposition yields a virtual-composite identity: the cross Gram matrix for shifts $a$ and $b$ acquires the previously known single-shift block evaluated at $a+b$. Along the physical curve $\\tau_q = \\frac{\\log q}{L}$, pairwise prime-power collisions therefore create walls at $L = \\frac{1}{2}\\log(qr)$. Grouping the quadratic terms by $n = qr$ gives a Dirichlet-convolution law with coefficient $\\frac{(\\Lambda * \\Lambda)(n)}{\\sqrt{n}}$, so the quadratic wall spectrum is supported exactly on integers with one or two distinct prime factors. Walls with two distinct prime factors are ghost walls: the quadratic Gram matrix changes although no new von Mangoldt layer enters. We prove that ghost-wall derivative jumps are rank one in each Legendre parity sector, whereas prime-power activation jumps have rank at most two; in the even sector every activation after $q = 2$ is strictly rank two and indefinite. For the split-residual Schur construction, we identify the second wall channel exactly with the endpoint error of a finite Legendre projection. This defect compresses to five adjacent Legendre modes in the even sector and seven in the odd sector and decays as $\\mathcal{O}(N^{-3/2})$ for fixed interior geometry. Using the Gelfond-Schneider theorem, we further prove that the full Schur wall jets at $q = 3$ and $q = 4$ are strictly rank two and indefinite in both parity sectors for every finite truncation $N \\ge 2$. The results are finite-scale structural statements and neither assume nor imply the Riemann Hypothesis.