Mathematical debugging ledger

• Field normalization. The choice $\Phi =a^{†}+a$ gives vacuum variance $\text{norm}{v}^2$, CCR $2\mathrm{i}\mathrm{Im}⟨v,w⟩$, and the number bound $4n+2$. A different convention changes these constants.

• Displacement compatibility. The targets are explicitly $o_j=b_j+2A(d,v_j)$. This avoids assuming that arbitrary targets come from one common coherent displacement.

• Unbounded products. Second moments are spectral quadratic moments. Commutator statements are made on a common core or as form identities, not as unjustified products of arbitrary trace-class and unbounded operators.

• Strong commutation. Vanishing symplectic pairings imply commutation of the entire one-parameter Weyl groups. This, rather than a formal core commutator alone, justifies the joint spectral law.

• Concentration criterion. Vanishing variance alone is insufficient. The exact condition is vanishing mean-square error about the prescribed target.

• Cutoff error. The exact tail probability is the state error only in the block-diagonal regime. Arbitrary coherence requires the square-root allowance. Failures are retained rather than renormalized away.

• Preparation metrics. Trace-distance closeness alone does not bound an unbounded field moment. Weighted preparation control or a tail certificate is required.

• Noise cross term. The cross term in (eq:noise-exact) is retained. Its removal requires centering with independence or a separate zero-cross-term proof.

• Witness error constants. Changing a channel affects both the coherent and dephased inputs; the resulting allowance is $2e_s$, not $e_s$. Preparation changes similarly contribute $2p$.

• Response ambiguity. The contact classification includes its scaling-degree restriction. Dropping that restriction can permit derivatives of ${\delta }_0$, which $F(0)=0$ does not all annihilate.

• Response units and sign. The source sign and $\hslash$ convention are fixed before evaluating the benchmark. The response pairing is not a probability.

• Environmental moment domains. A formal generator calculation does not by itself prove absolute continuity of an unbounded number moment. That property is part of the environmental certificate.

• Reduced algebra. A channel adjoint is not generally multiplicative. Preservation of the underlying enlarged CCR is not preservation of commutators by the reduced adjoint.

• Causal completeness. Unrecorded persistent bath memories or classical controls can invalidate a proposed local schedule. They must be included before computing ancestors.

• Joint endpoints. Input-independent marginals do not imply an input-independent joint channel.

• Statistical independence. The sample bound assumes independent attempted runs from a fixed family member. Adaptive dependence or repeated reuse of one preparation requires a different statistical argument.