Finite carriers, complete outputs, and access

The listed components of $B$ are an accounting requirement. They are not an assertion that an unregulated field theory admits the finite tensor factorization (eq:interface). A finite field-to-memory preparation, if physically claimed, must be supplied separately.

A sufficient mathematical preparation hypothesis is an isometry $J:C\to H_{\mathrm{in}}$, a fixed normalized pure ancillary state $|\Omega {⟩}_A$, and a unitary $U$ on the complete prepared space. After a fixed output identification,

A mixed fixed ancillary preparation can be purified only by retaining its purifier in the complete accounting. An input-dependent preparation must itself be specified as a code-to-complete-system map; it is not covered by assuming a fixed ancilla.

Finite spaces can be embedded in qubit registers with unused basis states. This observation is a mathematical encoding convention, not a physical field receiver or a gate-compilation certificate.