Relational Quantum State Selection: Geometry Reconstruction and Observer-Rooted Measurement

This preprint introduces event–relational physics (ERP), a finite constructive mathematical framework that addresses how a particular quantum state and measurement architecture can be selected rather than prescribed. Beginning from an inhabited completed relational occurrence, a compatible retained rooted frame, and one explicitly isolated physical postulate, ERP selects a Hermitian response-operator orbit, a positive Gram operator with an exact simple three-level spectrum, and a faithful finite quantum state. Relative to the retained rooted frame, the selected positive kernel and a finite relational geometry are mutually reconstructible through explicit forward and inverse maps. The same construction determines an observer-rooted projective measurement and calibration architecture. Standard positive-square-root, vectorization, purification, Schmidt, and modular constructions enter only after the particular quantum state has been selected; they are recovered downstream rather than claimed as new procedures. The proved result is finite and one-occurrence in scope and does not assert arbitrary POVMs, multipartite composition, continuum spacetime, gravitational dynamics, or a general derivation of quantum measurement. The record contains the complete manuscript and unified Supplemental Material. The supplement provides mathematical provenance, an explicit protocol-base witness certificate, technical details, application analyses, and clearly separated conjectural extensions. Supporting deterministic verification software and reproducibility materials are archived separately at https://doi.org/10.5281/zenodo.21741242.