Harmonic Structure of Spin-1 Quantum Measurement: Geometric Marginalization of Directional Amplitude Distributions

Abstract Recent work introduced a phase-coherent directional-amplitude framework in which an individual spin- particle carries a dipolar directional amplitude with a binary hemispherical response, while prepared ensembles are represented by continuous distributions of these individual amplitudes. Measurement probabilities arise through interaction with an externally oriented analyzer followed by geometric marginalization over the prepared directional-amplitude distribution. Here this framework is extended to spin 1 without introducing a primitive three-valued response. Each spin-1 particle is represented by two connected dipolar directional amplitudes, each retaining the binary hemispherical response of the spin- construction. Their four fine-grained binary combinations form the three spin-1 response classes . For an isotropic unprepared ensemble, requiring equal occupation of these three classes fixes the angular separation of the two directional amplitudes at The response of this fixed pair is then resolved into axisymmetric kernels . Matching their harmonic content to the standard spin-1 rotational laws determines the prepared directional-amplitude distributions. For preparation in , Corresponding distributions obtained for and . Their equal mixture recovers the isotropic distribution . Convolution of the prepared distribution with the fixed-pair response gives exactly for a subsequently rotated analyzer. The prepared distributions contain only the harmonic sectors required by spin 1. The component therefore emerges as second-rank structure of a connected pair of dipolar directional amplitudes rather than as an additional primitive directional amplitude. The same rotational transition probabilities are recovered independently by conventional coherent composition of two spin-½ amplitudes. The construction provides an explicit connection between fine-grained binary directional responses of an individual particle, preparation-dependent amplitude distributions across an ensemble, and the standard rotational statistics of spin-1 measurement.