Model specification: Local instruments and complete public information

An admitted protocol has finite-dimensional bipartite input $A\otimes B$, local quantum memories and local CP instruments, and classical public communication. It satisfies:

1. An initial standard Borel public variable $C_0$ is recorded and independent of the input and every untouched reference. Conditional on $C_0$, the private auxiliary resources of the two laboratories are independent of each other and of the input. Their preparation kernels are fixed when the input is changed. There is no other shared resource or preshared secret key.

2. Each instrument has a standard Borel outcome space, measurable conditional laws and updates, and trace-preserving total. Its choice depends only on its laboratory's earlier private history and the preceding public record.

3. After any explicitly supplied local aggregation, the genuine classical history has finitely or countably many coordinates in fixed standard Borel spaces. Aggregations retain all subsequently used private records and memories and are valid on arbitrary inputs with untouched finite references.

4. Every public message is generated by its sender from its own history and the preceding public record. An independent public-source draw uses a fixed kernel depending only on that record and acts trivially on quantum systems. Public controls, speaker choices, stopping decisions, and public randomness are included in the complete transcript $T$.

5. Unnormalized finite-prefix laws are one reference-compatible instrument family on arbitrary positive inputs. Tracing current memories and summing the next outcomes recovers the preceding reference matrix measure.

6. A terminal classical output is a measurable function of local history and $T$. If a terminal quantum readout is used after a complete infinite history, both its incoming local CP process and the readout instrument are supplied. They are local, reference-compatible, and jointly refine the complete-history matrix measure. Summing the terminal outcomes removes those refinements. An unspecified infinite quantum evolution is not declared to be a terminal instrument.

7. Each local output has a specified finite alphabet and a missing symbol $\perp$. A completed protocol produces its prescribed output at both laboratories almost surely on the actual input, with normalized total output. Completion on auxiliary inputs is not required.

Eve retains the original purification and the complete public transcript, not private instrument labels merely used inside a laboratory.