Theorem: Finite Enter spacetime: I-B-v1

Fix $0<\mu <2\pi$, $c>0$, a regular convex developing patch $U$, a clock dimension $D=2^q\ge 2$, and a payload dimension $d=2^p\ge 2$. Then the following finite operational package exists.

1. Its smooth background is the time-oriented $3+1$-dimensional Lorentzian manifold

   $$M={\mathbb{R}}_T\times {C_{\mu }}^{\circ }\times {\mathbb{R}}_z,g=-c^2\mathrm{d}{T}^2+\mathrm{d}{r}^2+{\alpha }^2{r}^2\mathrm{d}{\varphi }^2+\mathrm{d}{z}^2.$$

   Its completion has the declared conical worldsheet $\mathcal{W}$.

2. It has one designated live clock factor $C$, finite matter names, a location map, accessible algebras, and a complete finite control and resource interface.

3. Destructive clock Enter is exactly (eq:instrument), including every untouched reference. It consumes $C$, writes a classical code, and retains the subnormalized conditional matter block.

4. An explicit dictionary matches the code to geometric time labels and the surviving carrier names to three spatial rods. Exact equality with the actual reading-event coordinate holds on the event-calibrated support class of Proposition prop:fixed-event. Nontrivial multi-outcome history readout instead uses the separate event map $\kappa$.

5. Scheduled dependencies are contained in the prescribed metric causal relation. Actual intervention influence is contained in scheduled dependency.

6. Every prescribed straight future-null displacement inside the patch has the finite target-adapted SWAP protocol of Theorem thm:transport, with exact complete-channel correctness and the resource ledger (eq:peak). Its composition with clock readout is (eq:composed-channel).

7. The clock instrument has no universal left inverse on its full input space, and no finite canonical time commutator is available. The supplied metric and its time coordinate remain after clock consumption.

Signature, spatial chart, speed, lapse, calibration, and transport permissions are inputs. Einstein dynamics are not derived.

Proof. The geometric assertions are Propositions prop:geometry and prop:holonomy. Choose any finite set of sites in $U$, assign qubit registers, and prescribe the finite primitives of Definition def:experiment; this supplies item 2.

Theorem thm:instrument proves item 3. For item 4, choose distinct $t_k$ and a static sender $x_0$. The family

is a nonempty event-calibrated family. A classical task input selects which $e_k$ and preparation are used. For a pure matter input these are pure complete laboratory preparations. Thus actual event-time matching is exhibited, not merely postulated.

For a separate nontrivial history experiment, choose $\kappa (k)=(t_k,x_0)$, read at a later fixed event, and use the finite preparation (eq:history-isometry). Proposition prop:history gives all $D$ outcomes and their complete conditional outputs. The rod map is the explicit developing chart $\chi$.

Theorems thm:locality and thm:metric-containment give item 5. The sites (eq:chain-sites), their partition schedule, Theorem thm:transport, and Corollary cor:clock-transport give item 6. All preparations, clocks, site factors, records, and references have the declared ledger.

Propositions prop:no-return and prop:no-ccr prove item 7. The geometry is fixed metadata throughout the specification, so no instrument operation removes or changes its coordinate function. No source-extraction or metric-update map has been introduced.