Debugging ledger

The following corrections are required for a proof-ready implementation of the supplied plan.

• A complementary preparation is unnecessary. The proof uses separation and Jordan decomposition. No map ${\rho }_I\mapsto {\rho }_I\otimes \sigma$ is assumed.

• The local unit must be the ambient unit. The equality $f_{+}(1)=f_{-}(1)$ in the separation proof depends on it. Corners and fixed sectors must be reformulated with their own ambient unit.

• State richness is not optional. Restricted preparations establish membership in ${{\mathcal{D}}_{\mathcal{I}}}^{\perp }$, not automatically in $\mathfrak{A}(I)$. Even scalar-separating preparations need not supply all equal-snapshot fibers required by the theorem.

• Centers must not be discarded. Equal normalized sector states do not determine the sector probabilities. Complete central effects distinguish them.

• A block decomposition is not spatial factorization. The decomposition $M_2\oplus M_2$ describes a regional algebra. It does not identify complementary physical tensor factors.

• Exterior fixation is not algebra containment. Commutant membership and declared regional membership are different. The internal-device proof establishes membership by algebra multiplication and isotony.

• Global device extensions must be retained. The identity and ${\mathrm{Ad}}_F$ in Proposition prop:extensiongap have the same regional restriction but different cross-boundary actions.

• Arbitrary regional UCP maps need not be admissible. The sector-swap automorphism is incompatible with fixation of the shared exterior flux.

• Constraint compression needs invariant inputs. Multiplicativity of restriction was proved only for observables preserving the physical subspace.

• Complete records are required. The unital map is the complete recorded or nonselective channel. Selective normalization is not substituted for it.

• Joint outputs require a joint algebra. Separate one-region or one-record tests are not automatically a test of their full joint output.

• References are joint interfaces. An independent matrix reference is not a spatial complement. A charge-linked reference requires a joint constrained algebra.

• Expectations are extra structure. Gauge averaging is not automatically regional expectation. Trace-preserving expectation need not preserve a chosen state.

• Normal tests need normal closure. The normal theorem uses von Neumann subalgebras, not arbitrary norm-closed subalgebras tested only on normal states.

• Convergent numbers are not observable convergence. The bounded-sequence quotient and the exact refinement maps provide products and state pairings explicitly. An arbitrary convergent terminal law supplies neither.

• The refinement limit has a stated size. The subdivision family retains four physical dimensions. It is not an extensive field-theory scaling construction.