Spectral Channel Entropy Closure Hessians, Projected Degeneracy Volumes, and Entropy Channel Decomposition

Core Contribution Entropy channels are low-cost spectral sectors of closure-functional geometry. The projected closure Hessian determinant controls leading-order Gaussian channel volume. This paper develops the spectral structure of closure degeneracy. Paper 1 established the ontological-epistemic hierarchy: Negentropy -> Entropy Topology -> Entropy Paper 2 formalized entropy as closure degeneracy: S_C(c) = k_C log Omega_C(c). The present paper asks how closure degeneracy is internally organized. The answer is spectral. Near a closure-stable configuration, the closure functional admits a local quadratic expansion. Its second variation defines the closure Hessian. The eigenstructure of this Hessian measures local closure curvature. Low-eigenvalue directions are closure-soft; they support accessible relational variation and therefore contribute strongly to local degeneracy. Spectral projectors select these low-cost directions into entropy-channel subspaces. Gaussian integration shows that the volume of such a channel is controlled by the determinant of the projected Hessian: V_ch proportional to [det(P_Lambda H_C P_Lambda)]^(-1/2). The channel entropy is: S_ch = k_C log V_ch. Thus, up to constant scale terms: S_ch ~ -(k_C/2) log det(P_Lambda H_C P_Lambda). The central contribution is therefore: entropy channels are low-cost spectral sectors of closure-functional geometry. The previous paper defined entropy as the logarithmic measure of closure-preserving relational degeneracy. The present paper develops the spectral mechanism by which this degeneracy decomposes into entropy channels. Let C[phi] be a closure functional over relational configuration space R. Near a closure-stable configuration phi_0, the second variation of C defines the closure Hessian H_C(phi_0) = delta^2 C / delta phi^2 evaluated at phi_0. The eigenstructure of H_C measures local closure curvature. High-eigenvalue directions correspond to costly closure deformation, while low-eigenvalue directions correspond to closure-soft relational variation. Spectral projectors P_Lambda select low-eigenvalue subspaces. These projected subspaces define entropy channels when they form stable, accessible, or block-structured regions of low closure cost. Using a quadratic approximation around phi_0, the local projected channel volume is controlled by the determinant of the projected Hessian. When the projected Hessian decomposes into orthogonal invariant blocks, determinant factorization yields additive entropy-channel decomposition. This provides a spectral origin for entropy topology and prepares the candidate 1 + 3 + 8 minimal closure-channel structure. The paper also includes an appendix on the four projection topologies of entropy: Coherence-Diffusion Entropy, Observer-Mapping Entropy, Asymmetry-Induced Entropy, and Topological Coherence Entropy. These reinterpret an earlier fourfold map within the closure-degeneracy and spectral-channel framework. Keywords Spectral channel entropy; closure Hessian; entropy channels; spectral projectors; closure spectrum; low-eigenvalue plateaus; Gaussian channel volume; determinant factor; closure degeneracy; entropy topology; block projectors; channel decomposition; relational configuration space; closure functional; spectral entropy; projection topology.