Spectral Reconstruction of Hyperbolic Geometry as a Topological Invariant of a Self‑Adjoint Operator

<jats:p>We develop an operator‑spectral and K‑theoretic framework in which Lobachevskian hyperbolic geometry emerges directly from the spectral data of an unbounded self‑adjoint operator D on a separable Hilbert space H. No manifold, metric tensor, or geometric background is assumed. Geometry is reconstructed from spectral layers L(k) generated by projections P_k = E_D([λk, λ{k+1}]) or smooth localizations φ_k(D), together with boundary compression via an orthogonal χ‑boundary projection P_chi. On each boundary‑restricted layer L_chi(k) = P_chi L(k), we define an operator‑valued metric g_D(k) and a curvature functional K(k) = Tr(P_k [P_chi, [D, P_chi]] P_k), where the double commutator is trace‑class under the Hilbert–Schmidt condition [D, P_chi] ∈ S₂(H). The Hyperbolic Reconstruction Theorem shows that negative curvature (K(k) &lt; 0) is topologically quantized and equals an integer Fredholm index I(D, P_chi, k) = ind(P_chi P_k E_D+ P_k P_chi), coinciding with the Kasparov pairing ⟨[P_chi], [F_k]⟩. Hyperbolic curvature is thus an operator‑topological invariant. Using Weyl essential spectrum invariance and Atiyah–Singer index stability, we show that I(D, P_chi, k) resides in the Calkin algebra B(H)/S_∞(H) and is stable under compact perturbations D → D + K. Hyperbolic structure is therefore an essential spectral feature reconstructible from operator data alone. This mechanism also unifies nonsmooth geometric phenomena, linking spectral curvature, χ‑boundary tension, χ‑entanglement, χ‑chronal flow, and χ‑cosmology. Nonsmooth spacetime emerges as the projection Proj_chi(Ω_chi) of a zero‑entropy operator‑spectral substrate.</jats:p>