χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy

<jats:p>The χ‑operator geometry provides a new spectral framework for describing extremal black holes, whose zero‑temperature states exhibit non‑zero entropy and a collapse of surface gravity. In this work, we reconstruct the extremal horizon as a χ‑spectral degeneracy manifold and show that the vanishing of temperature corresponds to a structural reduction of the χ‑spectrum rather than thermodynamic triviality. The χ‑operator encodes the geometric, quantum, and gravitational degrees of freedom in a unified spectral form, allowing extremal black holes to be treated as zero‑temperature χ‑states with finite horizon complexity. This approach reveals a new mechanism for entropy retention in extremal configurations and establishes a purely operator‑geometric origin of extremal horizon structure.</jats:p>