Precise analytic acceptance tests

For any proposed kernel operations, Q3 must check:

1. Products. The component factors $u,v$ must satisfy the appropriate product condition; a standard sufficient condition excludes $(x,k)\in \mathrm{WF}(u)$ and $(x,-k)\in \mathrm{WF}(v)$ simultaneously Hormander (Theorem 8.2.10).

2. Pullbacks. A proposed restriction or composition $F^{\ast }u$ must avoid the normal set

   $$N_F=\{(F(x),\eta ):(\mathrm{d}{F}_x{)}^{\mathrm{T}}\eta =0\}$$

   in its input wavefront set Hormander (Theorem 8.2.4).

3. Pushforwards and tails. Properness on the support, or a separately proved nonproper convergence estimate, is required. Compact external smearing alone does not supply it.

4. Extensions. The actual extension locus, reference distribution, scaling bounds, normalization, and allowed corrections must be specified before an extension theorem is applied.

Local Wick and time-ordered-product existence results have their own local field hypotheses HWlocal,HWexistence. Their conclusions are not an automatic extension theorem for (eq:instantaneous) composed with the complete nonlocal lapse Hessian. Brunetti–Fredenhagen's finite-scaling-degree extension criterion concerns an already defined reference distribution and an actual scaling bound BF (Theorem 5.2); it does not create the absent reference product.

In particular, the phrase “extend across the light cone” is not one universal operation. Some positive-frequency products already exist by the wavefront criterion, whereas time-ordered or nonlocal characteristic compositions may need extensions. The locus and the operation must be identified separately.

Failure of a sufficient ordinary-product criterion is not a theorem that every renormalized or generalized realization fails.