Corrections required before a proof is possible

• A complex-linear DG equivalence is not automatically a quantum channel. Complete positivity, adjoints, density matrices, trace, and operational access are additional structures. They are not specified by presentability and complex linearity alone.

• The abelian example lies outside the printed connected-simple hypothesis. A torus or $\mathrm{GL}(1)$ is not a connected simple group in the sense of the supplied Family 069 statement. Replacing $G$ by a torus is a change of theorem, not a specialization justified by that statement. In particular, the printed determinant normalization involving $h^{\vee }$ cannot be substituted unchanged with $h^{\vee }=0$.

• An abelian local-system task is not a nondegenerate Whittaker task. A torus has trivial maximal unipotent subgroup. Our named task concerns actual finite-monodromy local systems on a fixed curve; it is not advertised as recovery of a nontrivial Whittaker module.

• The two meanings of free field are not identified. A chiral free-field realization of a $W$-algebra and a four-dimensional free massless scalar net are different objects. No dimension-changing or positive-net realization follows from the word “free.”

• Irrational categorical data do not establish unitarity or strong rationality. The source includes non-real levels. It does not provide the positive Hilbert realization needed by our operational task. The VOA–net comparison hypotheses discussed in Oai280-A are not silently imported Cheung280A.

• Sector labels do not authorize inter-sector quantum coherences. Our categorical state algebra is a finite direct sum. Arbitrary quantum coherence is allowed inside each multiplicity space, including correlations with a gauge factor and an external reference. Coherences between distinct superselection labels are not part of the declared task.

• Two stress probes are not process tomography. They cover a specified two-parameter conserved stress class. Complete channel coverage uses all input matrix units, or an equivalent complete Choi measurement family.

• A failed sufficient certificate is not an all-decoder exclusion. We distinguish exclusion of a tested decoder from bounds proved uniformly over every CPTP decoder allowed at the stated access.

• Global-phase and categorical coherence data require their own audit. Channel tomography cannot detect a common phase of a unitary implementation. Equality of implemented density-matrix channels does not by itself verify all natural isomorphisms, determinant twists, or central-torsor coherence.

• A marked-point configuration is not a spacetime boundary. We use a specified finite receiver interface. Removing a leg changes its channel and recovery capability, which is proved. We do not claim that moving marked points on the curve changes recovery, or that those points define a continuum entanglement wedge.