Nonuniform cells: an exact temporal proxy

Fix cell dimensions $s_1,\dots ,s_N\ge 2$ and a final core dimension $b_{\ast }\ge 1$. Let

Prepare a fixed isometry $J:C\to {\mathcal{H}}_{\mathrm{int}}$, choose one logical-input-independent unitary $U\in U(D)$, and then release the cells in order. Earlier outputs are untouched.

At cut $k$,

Haar statements refer to one Haar sample of $U$, not independently sampled states at different cuts. The pure input is fixed independently of $U$; $U$ is known when a decoder is invoked.

A finite conserving implementation may give each occupied memory and outgoing cell a common gap $\omega$, with a vacant zero-energy sector. The preparation occupies all memory cells. Extend $U$ on the all-occupied, degenerate internal sector and transfer each cell by SWAP. These operations commute with the declared sum of occupied-sector energies. This establishes a register conservation law, not a gravitational mass law Hawk05B,CheungHawk02B.

For integers $r,b\ge 1$, put

and define