Counterterms and the scope of the conclusion

The extension rule considered here is:

Keep the fixed Wick product on the old domain, admit sharp kernels continuously in the specified normal topology, and introduce no new counterterms on noncoincident characteristic relations.

The separated-support witness shows that a counterterm supported only on the ordinary observation diagonals

or on observation boundary faces cannot subtract the coefficient (eq:residue). The divergence persists when the three observation supports avoid all those sets.

There is an additional distinction worth making explicit. A distribution local in auxiliary Cauchy variables can acquire nonlocal support after Cauchy propagation. Such a modification is not automatically an ordinary observation-diagonal counterterm. Its pushforward, topology, and compatibility with the old Wick product would have to be specified separately.

More fundamentally, any subtraction performed on the regular approximants in (eq:Gdef) changes products that are already fixed in ${\mathcal{P}}_{\mathrm{mc}}$. Assigning a subtracted value only at the sharp kernel does not restore the continuity contradicted by Theorem thm:coefficient.

Thus the theorem excludes the stated normal-topology extension. It does not exclude a different boundary theory with additional renormalized operations, a stronger topology in which the approximants do not converge, or a smaller bulk algebra that omits $W_g$.