Prime-Uniform Cohomology and Extension Quivers for Two Symplectic Truncated Shear Towers

For an odd prime \( p\ge41 \), let \( H\cong(\mathbb Z/p)^2 \) be the elementary abelian Klingen unipotent quotient and \( L\cong\mathbb F_p^\times\times \mathrm{GL}_2(\mathbb F_p) \) the Klingen Levi group acting on two truncated polynomial shear towers \( P_{\le19}(19) \) (20 simple layers, dimension 210) and \( P_{\le39}(39) \) (40 simple layers, dimension 820) sharing the twist \( \chi=\chi_{(58,0,1)} \). We establish four main results that hold independently of any modulo-\( p^2 \) target-lattice descent: (1) an explicit divided-power Borel cohomology reduction determining the complete \( Ext^1_{k[H\rtimes L]} \) quiver across all \( 60\times60=3{,}600 \) ordered pairs of simple layers (Theorem 5.3) together with its general even-parameter extension law for \( m_0\ge2 \), \( p\ge2m_0+1 \) (Theorem 6.2), proving that cross-tower extensions vanish everywhere except at the collision prime \( p=2m_0+1 \) on the unique unordered pair \( \{(m_0-1,0),(2m_0-1,m_0)\} \), where a parameterized Bruhat--Steinberg 1-cocycle spans a one-dimensional class in both orientations; (2) a prime-uniform source-cohomology decomposition for the normalized leaf norm-line model, proving \( \operatorname{rank}_k\delta_p=4p-56 \) and \( H^1(H,J_{\rm norm})[p]\cong X_{20}\oplus Y_{40} \) for every legal prime \( p\ge41 \), with \( Y_{40} \) rotating into the diagonal of channels 1 and 3 at \( p=41 \) (Theorem 4.18); (3) an integral extension rigidity theorem and sharp orbit-invariant criterion (Theorems 4.4 and 4.8) combined with a first-principles parahoric point-orbit proof at p=41 establishing \( J_{\rm geom}\simeq J_{\rm norm} \) as \( \mathbb Z_{41}[H\rtimes L] \)-extensions and hence \( H^1(H,J_{\rm geom})[41]\cong X_{20}\oplus Y_{40} \) (Theorem 4.10}, Corollary 4.19, and Appendix B); and (4) a prime-uniform torus-weight census, rank-one coroot-collision polynomial adjoint identity \( B(B-(p-1)I)=pI \), power-rank decay law, and 45-stratum dimension formula (Theorems 3.1, 3.2, and 3.4). Target-side modulo-\( p^2 \) relation identities (\( p^2L_K=C_6+p^2\Lambda_\mu \)), transported target cocycles (\( g_Y+\Delta \)), and the dimension of the original geometric target fixed space \( V_{41}^H \) belong to a separate target-side study and are not claimed here.