Padding Transition

Abstract

Recoverable predecessors make the explicit last-tail transition an isometry.

Theorem 1.1 (Padding Transition).

$$\mathrm{conjTranspose}\left(W\left(a,head\right)\right)W\left(a,head\right)=1$$

Proof. Machine-checked in Lean as D5/S3/Quantum/StationaryPreparation/PaddingTransition.W_gram (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every finite alphabet, every multiset a and every chosen head, W(a,head) has orthonormal columns. No maximality or positive-capacity assumption is needed. Its memory is the sink together with each nonzero bounded tail and a bounded head index.

The probability law includes the one-tail boundary: positive head index emits head, while index zero emits the sole tail letter into the sink. Zero-probability letters cannot produce supported predecessors. On nonzero support, the emitted letter and successor recover the entire predecessor; this recovery discharges the off-diagonal Gram entries.

References

• Truth anchor: D5/S3/Quantum/StationaryPreparation/PaddingTransition.W_gram
• Dependency: D5/S3/Quantum/Entanglement/OccupancyWordSectors
• Dependency: D5/S3/Quantum/Entanglement/SequentialRegisterCircuit