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

Theorem: Admission of all three two-edge outputs

F4: admission by finite-degree reduction · lines 1021–1095 · proof included or linked

IX-Free@1#thm:treeadmission, IX-Free@1#eq:m1213, IX-Free@1#eq:a1213, IX-Free@1#eq:decomposition, IX-Free@1#eq:m1323, IX-Free@1#eq:a1323

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The functionals (eq:G1223)–(eq:G1323) belong to $P_{Σ}$ , including every metric admission test after configuration-direction differentiation.

Proof. The scalar-only output $G_{12, 23}$ has the compact scalar kernel already constructed. Its metric derivatives and their directional descendants vanish.

Consider $G = G_{12, 13}$ . Its polarized bilinear form is

b_{G} (h, k) = \frac{1}{2} ⟨ E_{12, 13; ν} [V], h_{00} (y) (L k)^{ν} (x^{'}) + k_{00} (y) (L h)^{ν} (x^{'}) ⟩ .

Thus $G (h) = b_{G} (h, h)$ and $G^{'}_{h} (ψ) [k] = 2 b_{G} (h, k)$ .

Define

m_{ψ} (y) a_{ψ, ν} (x^{'}) = 2 g (y) \int f (x) K (x, y) [\int f^{'} (x^{'}) S_{0 ν} (x, x^{'}) w^{ν} (x^{'}) d^{4} x^{'}] d^{4} x, = 2 f^{'} (x^{'}) \int f (x) S_{0 ν} (x, x^{'}) [\int g (y) K (x, y) h_{00} (y) d^{4} y] d^{4} x .

These formulas mean the corresponding proper distributional pushforwards. By Lemma lem:forest, both are smooth: they integrate the compact weighted tree except for a single vertex. They are compactly supported in $supp g$ and $supp f^{'}$ , respectively.

The metric derivative is therefore the actual linear kernel

G^{'}_{h} (ψ) = e_{00} m_{ψ} + L^{*} a_{ψ} = e_{00} m_{ψ} + q_{a_{ψ}} .

Its first summand is compact smooth and belongs to $M_{1}$ ; its second belongs to $Q_{Σ}$ .

For one configuration direction $ζ$ , the same formulas hold with $h$ replaced by $ζ_{h}$ and $w$ replaced by $L ζ_{h}$ . This is the configuration-independent metric derivative of $D_{ζ} G$ . Two directional derivatives of $G$ are constant functionals, whose metric derivative is zero. Higher derivatives vanish. Scalar directions have no effect. This checks every finite direction list.

For $G = G_{13, 23}$ , the corresponding smooth compact functions are

m_{ψ} (y) a_{ψ, μ} (x) = 2 g (y) \int f^{'} (x^{'}) K (x^{'}, y) [\int f (x) S_{μ 0} (x, x^{'}) w^{μ} (x) d^{4} x] d^{4} x^{'}, = 2 f (x) \int f^{'} (x^{'}) S_{μ 0} (x, x^{'}) [\int g (y) K (x^{'}, y) h_{00} (y) d^{4} y] d^{4} x^{'} .

Again its derivative is $e_{00} m_{ψ} + q_{a_{ψ}}$ , and the same finite-degree argument checks all descendants.

Displayed formulas in this passage

IX-Free@1#eq:m1213 · line 1044
$m_{ψ} (y) a_{ψ, ν} (x^{'}) = 2 g (y) \int f (x) K (x, y) [\int f^{'} (x^{'}) S_{0 ν} (x, x^{'}) w^{ν} (x^{'}) d^{4} x^{'}] d^{4} x, = 2 f^{'} (x^{'}) \int f (x) S_{0 ν} (x, x^{'}) [\int g (y) K (x, y) h_{00} (y) d^{4} y] d^{4} x .$
IX-Free@1#eq:a1213 · line 1044
$m_{ψ} (y) a_{ψ, ν} (x^{'}) = 2 g (y) \int f (x) K (x, y) [\int f^{'} (x^{'}) S_{0 ν} (x, x^{'}) w^{ν} (x^{'}) d^{4} x^{'}] d^{4} x, = 2 f^{'} (x^{'}) \int f (x) S_{0 ν} (x, x^{'}) [\int g (y) K (x, y) h_{00} (y) d^{4} y] d^{4} x .$
IX-Free@1#eq:decomposition · line 1063
$G^{'}_{h} (ψ) = e_{00} m_{ψ} + L^{*} a_{ψ} = e_{00} m_{ψ} + q_{a_{ψ}} .$
IX-Free@1#eq:m1323 · line 1081
$m_{ψ} (y) a_{ψ, μ} (x) = 2 g (y) \int f^{'} (x^{'}) K (x^{'}, y) [\int f (x) S_{μ 0} (x, x^{'}) w^{μ} (x) d^{4} x] d^{4} x^{'}, = 2 f (x) \int f^{'} (x^{'}) S_{μ 0} (x, x^{'}) [\int g (y) K (x^{'}, y) h_{00} (y) d^{4} y] d^{4} x^{'} .$
IX-Free@1#eq:a1323 · line 1081
$m_{ψ} (y) a_{ψ, μ} (x) = 2 g (y) \int f^{'} (x^{'}) K (x^{'}, y) [\int f (x) S_{μ 0} (x, x^{'}) w^{μ} (x) d^{4} x] d^{4} x^{'}, = 2 f (x) \int f^{'} (x^{'}) S_{μ 0} (x, x^{'}) [\int g (y) K (x^{'}, y) h_{00} (y) d^{4} y] d^{4} x^{'} .$
unlabeled · line 1034
$b_{G} (h, k) = \frac{1}{2} ⟨ E_{12, 13; ν} [V], h_{00} (y) (L k)^{ν} (x^{'}) + k_{00} (y) (L h)^{ν} (x^{'}) ⟩ .$

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section: F4: admission by finite-degree reduction