Abstract
Let \(\pi:\mathcal X\to\Delta\) be a projective one-parameter degeneration of complex threefolds whose central fiber \(X_0\) has finitely many ordinary double points \(\Sigma\), assume that \(\pi\) is locally a Morse smoothing at each node, and suppose that a finite group \(G\) acts algebraically and fiberwise. Building on the equivariant Hodge-atom assembly \(V_{\mathrm{van}}\simeq\bigoplus_{[p]\in\Sigma/G}\operatorname{Ind}_{G_p}^{G}V_p\), we develop an intrinsic local-to-global theory of attachment, global relations, and cohomological visibility. For each node \(p\), we prove \(L_{p,\mathrm{cost}}^H\simeq\mathbb Q^H(-2)\otimes V_p\) equivariantly, while \(E_p^H\simeq\operatorname{End}(V_p)\simeq\mathbb Q\) carries the trivial stabilizer representation. Hence \(W_{\mathrm{att}}^0\simeq\mathbb Q^H(-2)\otimes V_{\mathrm{van}}\), whereas \(\mathcal E_\Sigma^H\simeq\mathbb Q[\Sigma]\). A \(G\)-stable cycle--node datum determines the global relation map \(\Gamma_C^H=\lambda_{\mathrm{ODP}}^H\circ\iota_C\), its kernel, realized image, and unrealized quotient. For the four-nodal \(S_4\)-cubic, we compute \(R_{\mathrm{pair}}^H\simeq W_{2,2}\), obtain \(E_{\mathrm{geom,pair}}^H=E_\Sigma^H\), and prove \(\mathfrak c_{\mathrm{hyp}}^{\mathrm{att}}=0\).