Debugging the proposed formulation

The following qualifications are indispensable.

1. Conservation is not a complete Ward identity. A symmetric conserved traceless tensor gives conserved conformal-Killing currents. It does not, without a specified contact or local-charge argument, prove that these currents implement the intended transformations on every field.

2. Finite validation is not universal symmetry. A certificate on finitely many tests and matrix elements proves those tests and matrix elements. Extension to all tests requires coverage, continuity on a dense test family, or another independently proved completeness statement.

3. Local-to-global is an additional construction, not a universal impossibility. The correct conclusion is that this finite certificate does not supply a global action. Some theories, including the free massless scalar, have independently constructed conformal implementations HislopLongo,BGL. We do not assert that integration is impossible.

4. Global conformal geometry must be specified. Special conformal flows are not everywhere-defined flows on one Minkowski chart. A global formulation may require the conformal compactification or its covering space.

5. Charge preservation needs a locality or boundary argument. A total divergence does not automatically justify an unrestricted spatial integral of an operator-valued distribution. We prove local affine Ward-action preservation under an explicit spacelike-collar contract. Global charge equality remains conditional on the required limits.

6. Flat improvement is not a prohibition on gravitational coupling. An improved matter tensor can enter a separately specified curved background or gravitational theory. What is absent here is the identification and dynamical construction needed for that use CCJ,HollandsWald.

7. Deleting a scalar label is not necessarily deleting its descendants. Removing the name $Q=:{\varphi }^2:$ while retaining ${\partial }_{\mu }Q$ leaves the virial current available. A reduced field list must be examined explicitly.

8. Scaling dimensions are field labels, not automatically Hilbert-space energies. Finite scaling multiplicity is not a finite code dimension, a phase-space estimate, or an entropy history.

9. Cyclic and separating is not unique invariant vacuum. Modular standardness and uniqueness of an invariant vector are different assertions.

10. The En06-A theorem has an observable-limit contract. Its :algmicro uses product-preserving observable comparisons, not merely commuting channels CheungEn06A.

11. No ultraviolet identification is inferred. A conformal QFT can be relevant to an ultraviolet problem in an additional model. The Family 282 record does not itself construct an ultraviolet completion or matching theorem for Cheung's specified low-energy programme.

12. No reconstruction is hidden in our assumptions. The scalar $L$, its mixed products, and any required affiliated realizations are inputs to our local results. Their existence is not deduced from the Family 282 record.