Black Holes as Terminal Resonance Geometry in USP Field Theory: Finite Δf Saturation, Detector-Aware Diagnostics, and Horizon-Scale Tests

This document presents a v1. 4 upgrade of the USP Field Theory interpretation of black holes as terminal resonance geometries. In this framework, a black hole is not treated primarily as an infinite singularity, but as a finite resonance-saturated structure in which the mismatch field Δf (r) grows inward until ordinary atomic corridors fail, substructure decomposes, emission pathways are suppressed, and the core approaches a finite frequency ceiling. The central USP sequence is: |Δf (r) | increases inward → atomic failure → substructure decomposition → corridor suppression → finite saturated core A working finite radial resonance profile is introduced: f (r) = (fₘax · κ) / (fₘax r + κ) where κ is a geometry constant with units Hz·m, and fₘax is a finite saturation ceiling frequency. This formulation prevents singular divergence by replacing unbounded inward growth with finite high-frequency saturation. Atomic structure becomes unstable when the local resonance mismatch exceeds a binding tolerance: Δfbind = Ebind / h For hydrogen, this gives: Δfbind ≈ 3. 3 × 10¹5 Hz In an order-of-magnitude example for a 10⁶ solar-mass black hole, the radius at which the mismatch reaches the atomic binding threshold is: rbind ≈ 2. 9 × 10⁹ m which is comparable to the Schwarzschild radius for that mass scale. This provides an internal consistency check between horizon-scale geometry and the onset of atomic corridor failure. The model includes a minimal nonlinear elastic toy equation with boundary conditions and parameter anchoring. Finite saturation emerges from nonlinear stiffening, allowing the black-hole interior to be interpreted as a compressed resonance structure rather than a divergent singular point. Expanded in v1. 4, the paper introduces a structured emission model with three radial regimes: Core region, r < rₛat: saturated, non-emitting, effectively dark. Intermediate region, rₛat < r < rbind: atomic structure fails, leaving negligible direct emission. Transition layer, r ≈ rbind: atomic coherence is marginally maintained, producing partially suppressed emission. The emissivity cutoff is modeled as: j_ν (r) = j₀ exp−Δf (r) / Δfbind At the transition radius: Δf (rbind) ≈ Δfbind so: j_ν (rbind) ≈ 0. 37 j₀ This means the transition layer is not a hard emitting surface. It is a partially suppressed shell where atomic emission is already strongly reduced, while deeper regions become exponentially dimmed. A major v1. 4 addition is the detector-aware observation model. Horizon-scale images are treated not as direct photographs of the intrinsic resonance geometry, but as detector-filtered projections involving angular resolution, observing bandwidth, polarization channel, reconstruction method, and instrumental noise. The observed image is modeled as: Iₒbs (x, y;ν) = ∫ IUSP (x′, y′;ν) Kₒbs (x−x′, y−y′;ν) dx′dy′ + N (x, y;ν) where IUSP is the intrinsic USP emission profile, Kₒbs is the observational kernel, ν is the observing frequency, and N represents noise or reconstruction residuals. In this interpretation, the observed EHT-style ring arises from the overlap of: transition-layer emissivity near rbind, gravitational lensing amplification, resonance-locking opacity, and detector filtering through finite angular resolution and bandwidth. The dark interior is therefore interpreted as a resonance-accessibility effect: ordinary atomic emitters are absent, photon escape corridors are suppressed, and detector-accessible radiation is dominated by the outer transition layer. Opacity is modeled through resonance locking using a Lorentzian cross section: σ (f) = σ₀ Γ² / (f_γ, local − f (r) ) ² + Γ² where Γ is the effective linewidth, f_γ, local is the locally measured photon frequency, and f (r) is the local saturated resonance profile. Order-of-magnitude estimates suggest that mm-band observations, including EHT-style measurements near 230 GHz, may constrain Γ, absorber density nₐbs, and frequency-selective attenuation near the shadow boundary. The framework provides multiple observational diagnostics: Frequency-selective shadow attenuation. Weak frequency dependence of ring radius or ring width. Detector-bandwidth-dependent smoothing of shadow structure. Polarization scaling with shear index n. Jet opening-angle scaling, θⱼ ~ 1/n. Polarization-shear correlations. Small-impact-parameter lensing deviations from pure 1/r behavior. Finite-core signatures near saturated resonance regimes. Agreement with General Relativity is expected at large radii. Deviations are restricted to deep-interior saturation scales, near-transition emission layers, detector-resolved ring structure, or extremely small impact parameters. The model is disfavored if: No emissivity suppression is observed near horizon-scale transition regions. Ring radius, ring width, or edge sharpness show no frequency dependence across observing bands above declared sensitivity. Polarization and jet diagnostics fail to correlate with shear-based structure after controlling for plasma and reconstruction effects. No frequency-selective attenuation or resonance-locking signature is found within observational limits. Lensing remains strictly consistent with singular 1/r behavior at all accessible small-impact scales. This document is compatibility-first. It does not reject General Relativity or standard black-hole observations. Instead, it proposes a mechanism-level USP interpretation beneath the observed shadow, ring, opacity, polarization, and lensing structure, using finite Δf saturation and detector-aware diagnostics.