Temporal Reflection Separation Time

Abstract

A nondegenerate reflected spectral pair has canonical first-separation time one.

Theorem 1.1 (Reflected branches first separate at time one).

$$\begin{array}{c}\forall z:\mathbb{C},z\ne z^{-1}\Rightarrow \\ \mathrm{separationTime}(\mathrm{oneStepSpectralUpdate}(\mathrm{reflectedModes}(z)),\mathrm{modalSumReadout}(),(\mathrm{firstBranch}(),\mathrm{secondBranch}()))=1.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/FourierFibers/TemporalReflectionSeparationTime.reflected_branch_separation_time_eq_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

The reflected branch states collide at time zero and, when their reciprocal multipliers differ, separate at the first subsequent observation.

The proof instantiates the repository’s canonical separationTime and observedAt APIs; it does not introduce another break-depth definition.

References

• Truth anchor: D5/S3/ObserverMemory/FourierFibers/TemporalReflectionSeparationTime.reflected_branch_separation_time_eq_one
• Dependency: D5/S3/Observer/Separation/FiniteFutureCongruence
• Dependency: D5/S3/ObserverMemory/FourierFibers/SpectralFutureReadoutBridge
• Dependency: D5/S3/ObserverMemory/FourierFibers/TemporalReflectionBreakVisibility