Exact specification and reproduction checklist

The following finite procedure reproduces the ideal model without external code data.

1. Form the four $32\times 32$ Pauli matrices $g_i$.

2. Compute

   $$P={\prod _{i=1}}^4(I+g_i)/2.$$

3. Set the two columns of $V$ by (eq:encoder).

4. For each triple $B$, assemble $U_B$ using (eq:UB). Check ${U_B}^{†}{U}_B=I_8$.

5. Assemble $T=V/\sqrt{2}$ and $F=\sqrt{2}V$ with their printed index orientations.

6. Assemble $W$ using (eq:W-components). Check $W^{†}W=I_4$.

7. For each exterior triple $K$, assemble $U_K$ using (eq:UK).

8. Verify channel identities on every input matrix unit: four units for a qubit task and sixteen for the two-qubit bulk.

9. Build a decoder programme with all registers, arrivals, discards, instruments, and terminal replacements listed.

10. Freeze validation settings and failure allocations before collecting held-out shots.

11. Retain every trial record, including failures.

12. Compute the appropriate interval and apply the three-way decision rule.

Because $V$ has real entries fixed by a finite Pauli projector, all ideal identities above admit exact algebraic checking. Numerical checks are useful regression tests, but the proofs in the main text, not floating-point agreement, establish the identities.