Corrections to the proposed theorem architecture

The following corrections are necessary for the completed statement to be mathematically and operationally sound.

1. A clock basis must be declared. Computational-basis Enter realizes the intended observable only when the clock basis is already computational or a calibrated basis change is included.

2. Stationarity does not imply the desired evolution for an arbitrary observable. Sampled covariance is explicit in Definition def:covariance; Example ex:energy-basis supplies a counterexample without it.

3. Stationarity and a zero-energy constraint differ. Equations (eq:stationarity) and (eq:strong-constraint) are not interchangeable. The exact finite construction satisfies the stronger condition.

4. Uniform readings are a consequence in the exact covariant complete basis. One cannot simultaneously assume arbitrary clock weights and exact covariance of every basis projector for a stationary preparation.

5. Only the measured register's quantum output is absent. The complete process still returns the unmeasured system. Saying that the entire process has “only a classical output” would incorrectly discard its residual.

6. The Born attribution is conditional on the full predecessor class. Writing the instrument in Born form proves its algebraic law, not the predecessor's operational characterization. Hypothesis hyp:enter identifies the imported result and its applicability condition.

7. Normalized conditioning is not a linear channel. The instrument uses subnormalized blocks. Zero-probability branches have no prescribed normalized state.

8. Reference stability is an operator statement. Equality on one frozen state would not establish diamond equality. Equation (eq:exact-channel-identity) holds on every operator. Relational evolution of a reference-inclusive preparation additionally requires its own full stationarity.

9. A consumed clock cannot be measured again in the same continuation. The hybrid composition crosses the readout exactly once. Further physical readings need additional resources or a different instrument.

10. General circuit histories need not satisfy the additive constraint. Their propagation Hamiltonian (eq:propagation-H) is kept distinct from $H_C+H_S$.

11. Refinement uniqueness is class-relative. The ideal target, calibration, complete output interface, and local consistency estimates are frozen. Shared finite samples alone do not prove uniqueness.

12. No unsupplied theorem is silently imported. The XXI-A–C material available here is an outline-level comparison. The hybrid and telescoping results needed for En09-A are stated and proved above.