An explicit compressed cubic qubit

Use the witness of Theorem thm:cubicwitness, with oscillator vacuum, and define

The compressed cubic matrix is

Indeed, both diagonal cubic matrix elements vanish by scalar parity, and the off-diagonal element is $c_f>0$.

Use a one-qubit clock with two settings $f$ and $2f$:

Then

For encoded

the difference of conditional scored means is

The same two continuum preparations agree on all degree-at-most-two polynomial forms. Thus this is a complete finite readout of an explicitly compressed cubic distinction, not a claim that such a distinction is supplied by the external linear retarded field.