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

Lemma: Conditional polarized root representation

The root interface and the later compatibility result › Direct application versus polarization · lines 1362–1405 · proof included or linked

XVI-D@1#lem:root-polarization, XVI-D@1#eq:Q

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Suppose (eq:root-identity) holds on a real vector space containing the cosine and sine witness inputs and their required sums. With $f, g, ϕ$ and the root fixed, the real polarization

T_{D} (a, c; ϕ) = \frac{1}{2} (C_{3, D} [a + c, ϕ] - C_{3, D} [a, ϕ] - C_{3, D} [c, ϕ])

satisfies

T_{D} (a, c; ϕ) = \int_{M} a_{f, g} [ϕ] \cdot J_{D} (a, c) d^{4} y .

Its complex bilinear extension has witness value

T^{C} (γ_{1}^{λ}, γ_{2}^{λ}; ϕ) = Q_{λ}, Q_{λ} := \int_{M} a_{f, g} [ϕ]_{μν} S_{λ}^{μν} d^{4} y .

The integral defining $Q_{λ}$ exists whenever the stated compact smooth root is available.

Proof. Apply the diagonal root identity to $a + c, a, c$ and polarize. The common root is independent of the metric input, so linearity of the integral gives the polarized source.

Complex extension uses the same four-term expansion as (eq:BC). The complexified source evaluates to $S_{λ}$ by (eq:Slambda), yielding (eq:Q). Finally, the product of the compact smooth root with the ambient smooth source is compactly supported and integrable.

Displayed formulas in this passage

XVI-D@1#eq:Q · line 1382
$T^{C} (γ_{1}^{λ}, γ_{2}^{λ}; ϕ) = Q_{λ}, Q_{λ} := \int_{M} a_{f, g} [ϕ]_{μν} S_{λ}^{μν} d^{4} y .$
unlabeled · line 1368
$T_{D} (a, c; ϕ) = \frac{1}{2} (C_{3, D} [a + c, ϕ] - C_{3, D} [a, ϕ] - C_{3, D} [c, ϕ])$
unlabeled · line 1377
$T_{D} (a, c; ϕ) = \int_{M} a_{f, g} [ϕ] \cdot J_{D} (a, c) d^{4} y .$

Lemma: Conditional polarized root representation

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