Elementary proof checks

The following identities provide reproducible algebraic checks. They are not reports of executed numerical tests.

1. For $d=2$ and a Bell pair, the two-purifier Gram matrix has eigenvalues $3/2$ and $1/2$. Thus two unit-success Bell tests on independent purifiers exceed the permitted sum by $1/2$.

2. The benchmark matrix-unit complement is

   $${\mathcal{N}}^c(|i⟩⟨j|)={\delta }_{ij}|0⟩⟨0{|}_K.$$

   Checking only diagonal units would omit the coherence condition.

3. On the prepared slice, ${\Pi }_{\beta }V=V$ and $(I-{\Pi }_{\beta })V=0$.

4. The decoded $E$ channel is ${\tau }_F\otimes X_D$, so tracing $F$ returns every matrix unit $|i⟩⟨j{|}_D$.

5. For a Bell pair, exterior swap distillation changes the original proxy-pass probability from one to $1/4$, while preserving the message-reference state.

6. For qubit Pauli matrices, $[X,Z]=-2iY$ and ${\Vert [X,Z]\Vert }_{\infty }=2$. The sign does not affect the commutator obstruction.

7. The qutrit code in (eq:qutritcode) has constant single-register channels but orthogonal joint residual supports for distinct computational inputs.

8. Full-diamond allowances ${\epsilon }_t$ correspond to half-diamond allowances $e_t={\epsilon }_t/2$. Mixing the conventions changes the stated constants.