Manuscript En02-D@4 · En02-D-v4.tex · theorem · e19ec871

Theorem: Repeated dilation

Repeated emission and the loss-closed interface · lines 797–816 · proof included or linked

En02-D@4#thm:repeated

Each $W_{j}$ is isometric and energy preserving. Its coarse branch effect is $P_{j, n} (k) I_{n}$ , for every rail state and reference. With propagation,

H_{C_{t}} + H_{D_{\leq t}} + H_{F_{\leq t}} = E_{K} I

on the evolved initial sector.

Proof. Each extraction is isometric, distinct counts are orthogonal, and their effects sum to the identity. The branch dimension is $2^{n - k} 2^{k} = 2^{n}$ . Its energy is $E_{n - k} + E_{n} - E_{n - k} = E_{n}$ . Different sectors have different total energies. Compose with Proposition prop:scatter.

Displayed formulas in this passage

unlabeled · line 803
$H_{C_{t}} + H_{D_{\leq t}} + H_{F_{\leq t}} = E_{K} I$

Nearby in this section

section: Repeated emission and the loss-closed interface
proposition: Proposition: Propagation on the full loss-closed domain
section: Repeated emission and the loss-closed interface
section: Repeated emission and the loss-closed interface