Progressive Exterior Quantum-State Identifiability of Black Holes

We study a hierarchy of exterior measurements designed to separate black-hole response information from quantum-field-state information. Prior inverse problems can identify a restricted near-horizon response macrostate from active scattering channels, but response-kernel knowledge does not by itself determine the exterior quantum state. We therefore introduce finite-band outgoing wavepackets and ask which state directions become identifiable as spontaneous occupations, inter-packet coherences, anomalous pair correlations, and phase-sensitive quadratures are added. In a zero-mean two-mode Gaussian sector, the nullity of the observation map decreases as 8 -> 6 -> 4 -> 0, so complete covariance-matrix tomography is possible within that restricted class. This completeness is conditional: outside the Gaussian family we construct inequivalent states with identical first- and second-order data, distinguishable only by higher-order intensity correlations. More generally, for any fixed finite order K, distinct Fock-diagonal wavepacket states can be constructed with identical normally ordered factorial moments through order K and a difference first appearing at order K+1. The resulting no-go statement applies to the fixed finite hierarchy of normally ordered moments considered here, not to every conceivable finite quantum measurement. A Schwarzschild scalar benchmark based on direct numerical integration of the l=0 radial equation gives a greybody-filtered Unruh wavepacket occupation <N_f> = 0.02783535, stable under radial cutoff, horizon cutoff, and frequency-grid variations. A two-packet benchmark yields occupations 0.026306 and 0.024169 with a cross-packet coherence 7.23e-4. The restricted covariance inversion is structurally well conditioned, although the small coherence becomes noise-limited under a common absolute noise floor. We finally formulate a block inverse problem that appends finite-window QFT-state tomography to near-extremal response reconstruction. Richer exterior data progressively reveal more QFT-state structure; finite-mode Gaussian completeness can be achieved; fixed finite-order factorial-moment data remain incomplete on the enlarged Fock-diagonal family; and exterior-state completeness does not by itself imply a unique black-hole interior.