Theorem: Polarization, complexification, and witness matching

Under Inputs input:source–input:root:

1. ${\mathcal{O}}_{\mathrm{pol}}$ is symmetric in its tensor arguments.

2. It is real linear in the scalar.

3. The source, response, and diagonal completion have the degree-two homogeneity proved in Propositions prop:monomials and prop:Chom.

4. ${\mathcal{O}}_{\mathrm{pol}}$ is real bilinear in its tensor arguments.

5. Its unique complex-bilinear extension in the tensor arguments, with the scalar kept real, is

   $$\begin{array}{rl}{\mathcal{O}}_{\mathbb{C}}(a+\mathrm{i}b,c+\mathrm{i}d;\varphi )=& {\mathcal{O}}_{\mathrm{pol}}(a,c;\varphi )-{\mathcal{O}}_{\mathrm{pol}}(b,d;\varphi )\\ & +\mathrm{i}{\mathcal{O}}_{\mathrm{pol}}(a,d;\varphi )+\mathrm{i}{\mathcal{O}}_{\mathrm{pol}}(b,c;\varphi ).\end{array}$$

6. With the same data and domain conventions as revised XVI-E/F and current XVI-G, the specified witness pair satisfies, for each $\lambda >0$ and real smooth scalar evaluated there,

   $$\begin{array}{rl}{\mathcal{O}}_{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )& =T^{\mathbb{C}}({{\gamma }_1}^{\lambda },{{\gamma }_2}^{\lambda };\varphi )\\ & =Q_{\lambda }[\varphi ]=I_{\lambda }[\varphi ]\\ & ={\mathcal{A}}_1[f,g](R_{\lambda };\varphi ).\end{array}$$

7. On the common real response domain,

   $${\mathcal{O}}_{\mathrm{pol}}(a,c;\varphi )={\mathcal{A}}_1[f,g](B_{\mathrm{resp}}(a,c);\varphi ).$$

   The kernel construction in Section sec:kernel does not require this response-slot formula.

Proof. Symmetry follows from the definition. Scalar linearity follows from the scalar linearity of the three diagonal values. Homogeneity was proved above.

For bilinearity, Proposition prop:Chom gives the bilinear response polarization. Linearity of ${\mathcal{A}}_1$ on its stated common domain allows the finite response combination in (eq:Opol) to be taken inside ${\mathcal{A}}_1$, proving (eq:O-A) and real bilinearity. Alternatively, polarizing the root identity gives

which proves the same bilinearity from the root interface alone.

Real bilinearity verifies (eq:OC); complex bilinearity forces that formula in any extension, proving uniqueness.

For item 6, the definition of ${\mathcal{O}}_{\mathrm{pol}}$ is exactly the definition of $T_D$ in revised XVI-E, eq:TD, and revised XVI-F, eq:TD, on their common real domain. The polarization factor, diagonal insertion, source convention, and observation data are the same. Their four-term complex extension is exactly (eq:OC). Therefore ${\mathcal{O}}_{\mathbb{C}}=T^{\mathbb{C}}$ on that complexified domain.

The remaining equalities are the witness compatibility theorem of revised XVI-F, thm:compatibility, consolidated in revised XVI-E, thm:compatibility, and stated with the pointwise scalar-family quantifier in current XVI-G, prop:import XVIErev,XVIFrev,XVIG. Their operator and root-domain hypotheses are retained in the inputs of this note.