Remark: What must be certified separately

A transmission interval does not automatically certify:

• a physical gravitational identification;

• the implementation of the decoder;

• the accuracy of a phase correction;

• a common identification of residual wavefunctions across different experiments;

• a comparison of their physical complementary channels.

State residual bounds of the kind developed in Em04-A are useful interfaces, but a single-state bound is not automatically a referenced channel bound. A channel certificate must control the specified channel on arbitrary referenced inputs.

For an implemented phase error $\delta$, the noiseless decoded channel instead has the form

Its half-diamond error is at most

Indeed the relevant unitary channel has distance $|\sin \delta |$ from the identity: its numerical range has minimal modulus $|\cos \delta |$, and the pure referenced output distance is the square root of one minus the squared overlap. This phase allowance and any additional implementation error must be charged rather than absorbed into an unverified ideal-phase statement.