Spectral Future-Readout Bridge

Abstract

The finite spectral time-delay word is exactly the repository’s canonical future-readout word for diagonal modal transport.

Theorem 1.1 (Spectral delays reuse the canonical future word).

$$\begin{array}{c}\forall m,d,a:\\ \mathrm{futureReadoutWord}(\mathrm{oneStepSpectralUpdate}(m),\mathrm{modalSumReadout}(),d,a)=\mathrm{crystalTimeWord}(m,d,a).\end{array}$$

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

Source. Repository-derived.

Commentary.

For one-step diagonal spectral evolution and the modal-sum sensor, the repository’s canonical finite future-readout word equals the finite crystal time word coordinatewise.

This bridge prevents a second delay-coordinate API and connects finite Koopman-style time-delay reasoning to the existing observer-completion machinery.

References

• Truth anchor: D5/S3/ObserverMemory/FourierFibers/SpectralFutureReadoutBridge.future_readout_word_eq_crystal_time_word
• Dependency: D5/S3/ObserverMemory/FourierFibers/TimeShiftSpectralFiberTransport
• Dependency: D5/S3/ObserverMemory/Prediction/ConditionalEntropyStability