Few-Outcome Readout of Quantum-Fisher-Optimal Measurements: Sharp Bounds and Coherent-Synthesis Hardness

A quantum-Fisher-optimal measurement may resolve exponentially many eigenvectors even when its information-bearing score needs only a small outcome alphabet. This accepted private work derives support-aware and tail-adaptive few-outcome readout bounds, connects finite score spectra to a dense-spectrum quantization window, establishes Fisher-loss consequences of a coherent score-interface contract, and proves a faithful full-support hardness result for coherent selected-input synthesis. Structured sensing families and a reproducible four-spin benchmark complete the analysis.