 A semantic Gödel fixed point need not be canonical

Assuming PA is consistent, the theory $S=\mathrm{PA}+\neg {\mathrm{Con}}_{\mathrm{PA}}$ is consistent by second incompleteness. Its false axiom $G=\neg {\mathrm{Con}}_{\mathrm{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow \neg {\mathrm{Pr}}_S(┌G┐)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical Gödel sentence. A canonical Pi-1 construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.