Theorem: Complete buffered recorded channel

The preceding construction is a nonempty regular-part realization of the clock, five-command computation, and certificate-history transport.

For an input classical code word $x$, let $r(a,x)$ be its complete six-snapshot history together with the clock record, including rejection data on invalid branches. For an arbitrary operator $X$ on clock, code, source, and payload, define the canonical classical-interface blocks

The complete channel, with the canonical pinching of the classical input interface understood, is

Here ${\mathcal{U}}_{\mathrm{tr}}$ is identity on the named source factor and exact transport of the payload factor from $A$ to $B$; ${\omega }_{\mathrm{clean}}$ contains the final blank payload registers and zero work.

Equation (eq:complete-transport) holds with every untouched reference. Its geometric certificate interpretation holds branchwise on the stated promised canonical domains. Its causal dependencies are contained in the fixed host causal relation.

Proof. Theorem thm:instrument gives the clock blocks. On a classical code word, Theorem thm:reversible produces its six retained snapshots and clean work while acting as identity on $MP$. Copying an already classical word creates the displayed perfectly correlated record copies; it does not copy an unknown quantum state.

Lemma lem:swap proves exact payload transport, including its correlations with source and reference. All elementary operations are trace preserving on the complete interface; invalid branches are retained rather than postselected away. This proves the channel formula.

The semantic promise is used only to interpret the source-table entry. Proposition prop:branchpromise identifies the required branchwise class. For arbitrary off-diagonal code operators, the stated canonical classical-interface extension first pinches them. The coherent reversible circuit, before that classical extension, has its own isometric action and is not confused with the pinched map.

Every local operation has a positive duration on a regular station worldline. Every communication uses an explicit timelike axial segment in the convex tube, and each record is delivered before its next use. Theorem thm:physical applies.