Abstract

Algebraic reconstruction from a boundary region does not specify the resources needed to execute a recovery. We formulate observable readout, subsystem recovery, and instrument recovery as distinct finite-dimensional channel tasks, with correctness including arbitrary external references. Under specified Born semantics, we construct explicit decoders for the five-qubit perfect-code building block used in holographic tensor networks. For the accessible triple $\{1,2,3\}$, a nine-gate Clifford decoder uses three batch input qubits and no additional workspace. A streaming decoder uses two live qubits, two classical record bits, and no ancillary quantum register. An intrinsic uniqueness argument identifies the recovery channel on its entire three-qubit input space. Its Choi operator has a negative partial transpose across the first input, proving that one-live-qubit streaming recovery is impossible even with unrestricted classical memory. Thus the streaming optimum is exactly two, whereas batch access forces three. The same recovery channel has coherent environment dimension four; this is compatible with, but does not bound, instantaneous adaptive memory. For a rooted perfect-code tree, we prove batch peak $3^h$ and a streaming bound $h+1$, the latter optimal within a declared serial decoder-module class. A finite schedule recurrence prices that class without claiming unrestricted decoder optimality. We also prove a two-bit lower bound for a specified one-way distributed implementation and give a matching decoder. Half-diamond error budgets compose at the channel level. Exact Gaussian-integer checks and a finite schedule explorer are supplied as independently inspectable artefacts, not universal proof certificates. The results concern finite perfect-tensor code models, not an entanglement-wedge identification, spacetime emergence, or a gravitational memory bound.