Symmetry-Breaking Observability Gain

Abstract

Splitting an exact two-mode degeneracy turns a persistent hidden fiber into a faithful two-sample time readout.

Theorem 1.1 (Mode splitting increases observability).

$$\begin{array}{c}\forall u,v:\mathbb{C},u\ne v\Rightarrow \\ \neg \mathrm{Injective}(a\mapsto \mathrm{crystalTimeSample}(\mathrm{degenerateModes}(u),a))\wedge \\ \mathrm{Injective}(\mathrm{firstCrystalTimeWindow}(\mathrm{splitModes}(u,v))).\end{array}$$

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

Source. Repository-derived.

Commentary.

An exactly degenerate two-mode system has a nontrivial all-time hidden direction, whereas distinct split multipliers make the first two time samples injective.

The theorem captures an information gain caused by lifting spectral degeneracy. It is a finite observer statement and does not assign a physical mechanism to the split.

References

• Truth anchor: D5/S3/ObserverMemory/FourierFibers/SymmetryBreakingObservabilityGain.symmetry_breaking_observability_gain
• Dependency: D5/S3/ObserverMemory/FourierFibers/DegenerateModeHiddenFiber
• Dependency: D5/S3/ObserverMemory/FourierFibers/TemporalReflectionBreakVisibility