Multiscale Visibility Theory: A Task-Relative Operator-Geometric Framework for Observation, Robustness, and Detectability

Multiscale Visibility Theory (MVT) is introduced as a task-relative operator-geometric framework for determining how prescribed information survives and remains accessible through a structured observation architecture. Rather than evaluating observation quality by transform magnitude, output energy, rank, or signal prominence alone, MVT measures the accessibility of a specified task direction or task subspace through the row space of the composite operator A_A = R_E W_rho H_beta P_Omega. Unlike projection-based diagnostic frameworks centered on discrimination between nominal and fault-related system behavior, MVT tracks an independently prescribed task as observation architecture, scale, representation, and retained access vary. The formulation unifies exact stagewise retention, task-subspace preservation, scale-indexed visibility and stability, worst-case erasure robustness, collective visibility, covariance-normalized statistical separation, and decision sufficiency while keeping geometric accessibility, statistical detectability, and decision adequacy mathematically distinct. A reproducible MATLAB implementation verifies the analytical relations using a controlled synthetic benchmark with a weak 260-Hz prescribed task and a stronger 380-Hz nuisance. Within the same frequency-indexed observation family, task visibility is maximized at 260 Hz, whereas complete-signal observation energy is maximized at 380 Hz. At the nominal noise level, D_max = 18 > Gamma_D = 8.563847350668, yielding a decision-visible interval of 258.5-261.5 Hz. A four-row phase-redundant observer preserves full task-model visibility through two row erasures, while two individually subthreshold branches at 258 and 262 Hz become jointly decision-visible with D_12 = 10.470833520729. All 12 numerical audits satisfy the 10^-10 tolerance.