Projective return
A half-turn preserves projective scope while changing the nonzero local carrier state.
Four Dual Forms · F 03
Time Scope
Local phase and projective return
Abstract
Time carries local phase through physical distance and duration. Scope reads the direction of a state through its projective identity. A half-turn changes every nonzero local state while leaving its projective scope unchanged. Linear polarization and a two-level quantum state carry the same double-angle relation.
Uπz = −z
Scope(Uπz) = Scope(z)
(cos 2α, sin 2α)
Record
Shared verifier
The shared finite verifier carries the phase and projective relations beside the common root.
Stated realizations
Polarization and Pauli two-level systems are the declared realization domains; the page makes no apparatus-specific new-physics claim.
Scope
Claim boundary. The exact result is the phase and projective-scope relation on V× = ℝ² ∖ {0}, with the stated optical and two-level realizations. An apparatus, calibration, and independent measurement remain their own record.