Automatic differentiation computes loss Hessians, while interpreting inter-layer interactions requires an explicit choice of graph-indexed blocks. This paper presents an analytical formalism that decomposes the full Hessian into blocks indexed by the DAG of an architecture. The standard decomposition $H = H^{GN} + H^T$ separates the Gauss-Newton component (positive semidefinite for a convex output loss) from the tensor component (residual curvature). If every node map is locally affine in its parent activations, the tensor component of the input Hessian vanishes on each smooth cell ($H^{T}_{v,w}=0$, $H^f_{v,w}=H^{GN}_{v,w}$); ReLU activations alone do not suffice. The full parametric Hessian contains residual terms that do not reduce to the GGN. Building on this decomposition, we use diagnostic metrics (inter-layer resonance $\mathcal{R}$, geometric coupling $\mathcal{C}$, stable rank $\mathcal{D}$, GN-Gap) with stochastic HVP-based estimates whose cost depends on graph evaluation and the number of probes. The theoretical analysis gives conditional bounds on curvature decay and residual-chain transport; empirical examples span fully connected MLPs (Exp. 1-5) and convolutional architectures (ResNet-18, ${\sim}11$M parameters, Exp. 6). When the architecture reduces to a single node, all definitions collapse to the standard Hessian $\nabla^2_θ\mathcal{L}(θ)\in\mathbb{R}^{p\times p}$. Archival correction. The original general novelty claims are no longer maintained: general DAG-Hessian calculus has prior precedents. This is a corrected archival version.
Evaluating block-diagonal curvature models involves two questions: how much curvature they omit, and what a specific correction changes at a finite step. We measure the squared off-block fraction of the parameter Hessian against references for group sizes and the observed within-group curvature profile. In CIFAR-100 ResNet-18 and a skip-free counterpart, similar initial discarded fractions accompany different profiles. At saved trained ResNet-18 weights, changing from stage to layer groups moves estimated off-block mass from below to above the profile envelope; an approximate conditional screen resolves this comparison in all six seed/input cells, while some skip free comparisons remain unresolved. Separately, in saved ResNet-50/Stanford Cars states, we compare candidates formed from off-group Hessian action, the remaining model residual and their combination through the same Hessian-informed map. At unit scale off-group input alone raises native minibatch loss relative to the base preconditioner candidate in 17 of 18 state/geometry comparisons, yet the combined candidate has lower loss than the remainder-only candidate in nine of those cases; preferences vary with scale and correction-norm matching. The profiles help assess block models and alternative groupings, while same-state candidate comparisons test a particular correction without equating structural mass with training gain.