Operator–Spectral χ‑Entropy and Hyperbolic Vacuum Structuring: Statistical Foundations of χ‑Reality

<jats:p>χ‑entropy is introduced as a rigorous operator–spectral invariant quantifying the internal differentiation of χ‑structures on the χ‑Boundary. The χ‑entropy functional S_chi = -Tr(Delta_chi * ln(Delta_chi)), defined via the χ‑time operator Delta_chi = exp(-T_chi) and the χ‑tension operator T_chi, arises directly from the non-local spectral action of the χ‑boundary derivation B = D * P_chi - P_chi * D, where D is an unbounded self-adjoint operator on a Hilbert space H and P_chi is an orthogonal projection onto the χ‑Boundary. Hyperbolic spectral curvature K(k) &lt; 0 induces non-compact su(1,1) symmetry on spectral layers L(k) = P_k(H) and determines the non-equilibrium statistical distribution of χ‑vacuum modes. A complete operator–spectral reconstruction of χ‑entropy is derived, establishing its invariance under χ‑stable trace-class perturbations [V, P_chi] in S_1, its exact scaling under unitary representations, and its governing role in χ‑collapse dynamics, sub-harmonic χ‑fluctuations, and emergent χ‑noise. These results demonstrate that χ‑entropy forms the statistical backbone of χ‑reality, unifying non-commutative spectral geometry, non-compact representation theory, and quantum statistical mechanics within the non-perturbative ZEBTS‑SUPER 9.0 architecture.</jats:p>