Theorem: Payload naturality and complete instrument preservation

For every payload assignment on the fixed skeleton,

The identity holds after tensoring both sides with the identity on every finite-dimensional untouched reference.

Matched payload modifications use exactly the same routes, residues, schedule, encodings, and decodings.

Proof. Follow a topological ordering of the logical network. Every access or route operation moves a typed factor by swaps with reserved blank factors. On the logical factors this is the specified identity wire under tensor-factor relabelling.

When the inputs of a payload vertex have arrived, its original complete map is applied to precisely those factors. In a macrovertex, the original finite zero-duration subnetwork is applied in a topological order. This has the same contraction as the source subnetwork because operations on independent factors commute.

Induction over the vertices therefore gives the same complete map on all logical wires and records. Discarding unused blanks and identifying external ports gives (eq:logicalidentity). The argument is an equality of linear maps, so its tensor extension by ${\mathrm{id}}_R$ is immediate.

The schedule was constructed without inspecting payload maps. Replacing any specified complete payload map leaves every routing and comparison interface unchanged.