Observer Quality as a Resource Constraint in Quantum Darwinism: Optimal Decoding, ε-Approximate Spectrum Broadcast Structure, and a Central-Spin Worked Example

Quantum Darwinism (QD) and Spectrum Broadcast Structure (SBS) formalize a mechanism by which many observers can learn the same classical pointer value of a system by measuring different fragments of its environment. Most QD analyses treat the observer as idealized: unlimited access, perfect calibration, and no temporal constraints. This paper introduces an explicit observer quality parameterization and propagates it through redundancy and objectivity statements in a way that remains stable under an explicit ε-SBS trace-distance error model. Our main technical contributions are: (i) a decoder-level, tight (Chernoff-optimal) sample-complexity characterization for the number of environmental fragments needed to infer a pointer value under a calibrated observer model, (ii) a data-processing theorem showing that calibration noise (modeled as a CPTP channel) degrades the quantum Chernoff exponent, and (iii) an ε-robustness theorem upgrading ideal-SBS sample-complexity bounds to approximate-SBS states with additive error control. We ground the formalism in a fully worked open-system example: a central-spin pure-dephasing model whose conditional fragment states can be computed analytically, yielding explicit formulas for the observer parameters (RO, CO, τO) as functions of couplings, readout noise, and acquisition time. We also highlight a sharp performance gap between collective (coherent) decoding and product-measurement decoding for pure-state records, directly linking quantum memory horizon to redundancy. Finally, to connect this physics formalism to the DLN observer-quality framework, we introduce a three-state stage index qO ∈ qD, qL, qN mapping Dot/Linear/Network stages to memoryless, product, and collective decoding classes. We then prove three dynamical results about observer topology over time: (A) a dynamical redundancy theorem showing that the long-run effective Chernoff exponent depends on the observer's revision topology RO, with full-cycle revision achieving the adaptive optimum while expand-only revision collapses permanently to the linear baseline; (B) an "inverted sophistication" theorem showing that an unmonitored collective decoder can be strictly worse than optimal product decoding below a critical coherence fraction; and (C) a pointer-accessibility proposition formalizing how the DLN stage determines which pointer distinctions are resolvable from a fixed fragment budget, providing an operational basis for observer-designed control of accessible quantum outcomes.