Abstract

We give a finite-dimensional certification framework for a specified decoder acting on a declared accessible interface. The target may be a logical code, a subsystem, or a finite direct-sum algebra. Error is half-diamond distance, so the claim includes preservation of correlations with independent references. We prove the required task separations, the full-code constant-complement criterion, a programme-to-channel theorem with explicit resource ledgers, and dimension-correct Choi-to-channel bounds. A five-qubit stabilizer tensor supplies a fully specified connected two-cell encoding. Its normalization, subsystem decoders, hybrid-algebra decoder, and several optimal access errors are proved locally. For a three-share regression task we also prove decoder uniqueness, a minimal coherent environment dimension, an exact streaming peak of two qubits, and a four-message lower bound in a specified one-way model. General mixed-target tomography and a qubit Bell-correlation procedure give simultaneous finite-confidence intervals for complete implemented channels. Calibration, access assumptions, transfer allowances, and all-decoder obstructions are kept separate. Explicit noise examples and shot-count calculations identify both useful certificates and certificate-specific limitations. No gravitational dictionary, continuum geometry, or device-independent certification is asserted.