The Physical Implications of Euler's Equation: Amplitude–Phase Locking via Spacetime Safety Equations

Preprint. Publicly released on Zenodo prior to journal submission.This revised note reinterprets Euler’s formula eiθ=cos⁡θ+isin⁡θeiθ=cosθ+isinθ=cosθ+isinθ as an operational bridge between (i) shielded complex-phase rotation and (ii) real-valued reported observables obtained via Hermitian projection. The central aim is to clarify how unitary time evolution can be treated as phase routing under limited time readout resolution, in a way that is compatible with the Blackfield / Spacetime Safety program. Compared to v1, v2 tightens the operational definitions (phase driver →→→ SU(2) routing →→→ projection) and updates the related-work linkage to the current Spacetime Safety / readout-layer / rank-one representation trilogy. The discussion is aligned with the readout-layer viewpoint: the imaginary unit is not merely representational, but functions as a rotation generator whose effects become measurable only through the declared observation/projection rules and associated rescaling conventions. Related preprints: doi:10.5281/zenodo.18158453; doi:10.5281/zenodo.18169180; doi:10.5281/zenodo.18169038.