Manuscript IX-C-ST@1 · IX-C-ST.tex · proposition · 892f50c4

Proposition: Degree-two relation kernels

Concrete relations and evaluated scalar descent · lines 1409–1432 · proof included or linked

IX-C-ST@1#prop:relations, IX-C-ST@1#eq:qrelation

The exact relation equations are

q_{L} (v) = 0 q_{R} (v_{R}) = 0 ⟺ \partial_{t} (v K) = 0, ⟺ \partial_{t^{'}} (v_{R} (y, x^{'}) K (x^{'}, y)) = 0.

Proof. Observation-variable integration by parts gives

q_{L} (v) = 2 ⟨ \partial_{t} (v K), ϕ (x) h_{00} (y) ⟩ .

There is no boundary term because $v$ is compactly supported inside $V^{2}$ . The independent scalar and metric slots recover the coefficient on all product tests. Product-test density identifies the joint distribution. The reverse calculation is identical.

Displayed formulas in this passage

IX-C-ST@1#eq:qrelation · line 1412
$q_{L} (v) = 0 q_{R} (v_{R}) = 0 ⟺ \partial_{t} (v K) = 0, ⟺ \partial_{t^{'}} (v_{R} (y, x^{'}) K (x^{'}, y)) = 0.$
unlabeled · line 1424
$q_{L} (v) = 2 ⟨ \partial_{t} (v K), ϕ (x) h_{00} (y) ⟩ .$

Proposition: Degree-two relation kernels

Displayed formulas in this passage

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