Phase-aware tableau measurement agrees with the projective instrument

If some row anticommutes with $M$, [F1] gives probability $1\mathrm{/}2$ to each bit. For either sampled bit, [F2] proves the updated rows are a rank-$n$ stabilizer for the normalized state obtained by applying $P_b$. Thus the tableau state is the ray of that branch vector. [F6] defines the instrument's normalized branch density, and [F7] identifies it with the vector's rank-one projector; [F8] makes that projector independent of the chosen unit representative. Therefore the tableau and instrument have the same two branch states and probabilities.

If every row commutes with $M$, [F3] gives one deterministic outcome and [F4] computes its bit from the signed row product. The post-measurement state for that bit is unchanged because it is already in the eigenspace. Facts F6, F7, and F8 identify its probability-one branch density with the rank-one projector of the same ray; [F5] says the other branch is impossible and has no normalized state. Hence the tableau output agrees with the instrument in this case as well.

Every Hermitian Pauli either commutes or anticommutes with each stabilizer row, so the two cases above exhaust all measurements. Therefore the tableau rule agrees with every defined branch of the projective instrument, including its zero-probability boundary.