Manuscript IX-Free@1 · IX-Free.tex · section · 33b20e96

The individual outputs

F3: construction of the three two-edge distributions › The individual outputs · lines 951–995

IX-Free@1#eq:G1223, IX-Free@1#eq:G1213, IX-Free@1#eq:G1323

Let $π_{ab}$ denote projection onto the indicated vertices. The outputs are the actual quadratic functionals

G_{12, 23} G_{12, 13} G_{13, 23} = ⟨ E_{12, 23} [V], \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = ⟨ E_{12, 13; ν} [V], h_{00} (y) w^{ν} (x^{'}) ⟩, = ⟨ E_{13, 23; μ} [V], w^{μ} (x) h_{00} (y) ⟩ .

For (eq:G1223), the compact scalar coefficient kernel is

Sym_{x, x^{'}} \partial_{t} \partial_{t^{'}} (π_{13})_{*} E_{12, 23} [V] .

There is no sign change from two scalar integrations by parts.

For (eq:G1213), first form the compact residual kernel

R_{ν} (y, x^{'}) = (π_{23})_{*} E_{12, 13; ν} [V],

then transpose $L$ in its second variable and symmetrize the two metric slots. For (eq:G1323), use

R_{μ} (x, y) = (π_{12})_{*} E_{13, 23; μ} [V]

and transpose $L$ in its first variable.

These transpositions are defined on compact distributions. Indeed, applying $L$ to a smooth joint test function in one variable is continuous in the joint smooth topology. Its transpose therefore acts on compact distributions. Finite propagation supplies a common compact field support. This construction does not complete any presentation space or invoke a quotient topology.

Displayed formulas in this passage

IX-Free@1#eq:G1223 · line 955
$G_{12, 23} G_{12, 13} G_{13, 23} = ⟨ E_{12, 23} [V], \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = ⟨ E_{12, 13; ν} [V], h_{00} (y) w^{ν} (x^{'}) ⟩, = ⟨ E_{13, 23; μ} [V], w^{μ} (x) h_{00} (y) ⟩ .$
IX-Free@1#eq:G1213 · line 955
$G_{12, 23} G_{12, 13} G_{13, 23} = ⟨ E_{12, 23} [V], \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = ⟨ E_{12, 13; ν} [V], h_{00} (y) w^{ν} (x^{'}) ⟩, = ⟨ E_{13, 23; μ} [V], w^{μ} (x) h_{00} (y) ⟩ .$
IX-Free@1#eq:G1323 · line 955
$G_{12, 23} G_{12, 13} G_{13, 23} = ⟨ E_{12, 23} [V], \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = ⟨ E_{12, 13; ν} [V], h_{00} (y) w^{ν} (x^{'}) ⟩, = ⟨ E_{13, 23; μ} [V], w^{μ} (x) h_{00} (y) ⟩ .$
unlabeled · line 971
$Sym_{x, x^{'}} \partial_{t} \partial_{t^{'}} (π_{13})_{*} E_{12, 23} [V] .$
unlabeled · line 978
$R_{ν} (y, x^{'}) = (π_{23})_{*} E_{12, 13; ν} [V],$
unlabeled · line 983
$R_{μ} (x, y) = (π_{12})_{*} E_{13, 23; μ} [V]$

The individual outputs

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