Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a

Response theorem and class boundary

Response theorem and class boundary · lines 1498–1527

The theorem is about smooth responses and explicitly specified completion routes. It is not a construction of a compact observable kernel.

In particular, a response tensor $U_{D}$ is not a kernel $w_{D}$ on two metric slots and a scalar slot. If such a compact kernel is later constructed, comparison with the frozen XI-G kernel $w_{0}$ must use the relevant free-shell null space

I_{sol} = {z : ⟨ z, γ_{1} \otimes γ_{2} \otimes ϕ ⟩ = 0 for all smooth □ γ_{a} = 0, (□ + m^{2}) ϕ = 0}

within the specified compact kernel space.

The condition $w_{D} - w_{0} \in I_{sol}$ transfers the frozen obstruction. Changing a source does not establish or exclude that condition. A different-class proof would not itself prove admission XVIA,XVIB,XIG.

The separate characteristic quantum-product programme described in the supplied coverage material is not discharged by these smooth witness-response calculations.

Displayed formulas in this passage

unlabeled · line 1509
$I_{sol} = {z : ⟨ z, γ_{1} \otimes γ_{2} \otimes ϕ ⟩ = 0 for all smooth □ γ_{a} = 0, (□ + m^{2}) ϕ = 0}$

Nearby in this section

theorem: Theorem: Prepared witness-response package