Theorem: Oai280-B connected finite reconstruction result

For the printed encoding and the declared finite interfaces:

1. every ideal boundary set of at least three qubits has a constructed complete accessible decoder with zero logical error;

2. every ideal boundary set of at most two qubits has optimal full-code half-diamond error $3/4$;

3. the named ordered triple has a two-slot exact decoder and a sharp one-slot error floor $1/2$ in Assumption ass:stream;

4. complete implemented region–decoder channels satisfying the access and validation contracts obey the simultaneous finite-data guarantee (eq:falseaccept);

5. independently justified model and transfer bounds may be composed without identifying them with statistical uncertainty;

6. the two-pentagon patch is an isometry with the constructive subsystem recovery rules of Theorem thm:patchrecovery;

7. arbitrary-referenced finite-factor subsystem recovery is certifiable under Theorem thm:subsystemcertificate.

These conclusions are finite code and channel statements. They do not establish a gravitational duality, an entanglement wedge, a metric, Einstein dynamics, a Page curve, or an inherited characteristic quantum coefficient.

Proof. Items one and two are Theorem thm:idealaccess. Item three combines Proposition prop:streamdecoder and Theorem thm:oneslot. Item four is Theorem thm:main applied to (eq:implemented); its access meaning uses Proposition prop:accessfactor. Item five is Proposition prop:modelcomparison and the triangle-inequality transfer step. Items six and seven are the named patch and subsystem theorems.

The proved constructions contain finite matrices, channels, register access, and statistical contracts. No map identifying these objects with gravitational observables or with the unconstructed characteristic coefficient is among their hypotheses or conclusions.