Abstract

We specify a fixed-realization interacting SYK source coupled to a continuum of free scalar reservoir modes with declared infrared and ultraviolet cutoffs. A normalized Euclidean-filtered pure source state is prepared before a logical diary arrives. A single message-independent, parity-preserving swap loads an even-parity qubit diary into two source modes. The displaced carrier is retained in the complete residual, including its correlations with the remaining source.

We construct the self-adjoint continuum Hamiltonian, determine the complete code isometry, and derive a reference-preserving finite approximation for every prescribed finite packet algebra and finite observation interval. If $G$ is an explicit interaction bound, $m$ is a total reservoir occupation cutoff, and $\Delta$ is a frequency-mesh bound, the approximation allowance is

The finite approximation supplies all logical matrix units and both channel outputs. A complementary Choi-state certificate ${\hat{e}}_J$ gives

with the code dimension factor retained. Independently, a specified finite CPTP decoder can be certified directly on its normalized Choi operator. We prove the exact microscopic energy ledger, an obstruction for commuting source couplings, and continuity requirements for an information endpoint.

The manuscript establishes a coherent microscopic interface and a quantitative certification procedure. It does not exhibit a small dynamical residual certificate for the interacting model, derive a gravitational geometry from that model, or establish a gravitational evaporation endpoint. Those missing results are identified separately rather than assumed inside a purported evaporation theorem.