Manuscript IX-Free-2@1 · IX-Free-2.tex · assumption · abb2397a

Assumption: Locked stages and inherited graph outputs

Release alternatives and inherited contract › Explicit inherited premises · lines 410–490

IX-Free-2@1#ass:graphs, IX-Free-2@1#eq:firstL, IX-Free-2@1#eq:firstR, IX-Free-2@1#eq:oneedges, IX-Free-2@1#eq:G1223, IX-Free-2@1#eq:G1213, IX-Free-2@1#eq:G1323, IX-Free-2@1#eq:degree2L, IX-Free-2@1#eq:degree2R

Back to manuscript

The fixed mixed kernel $K$ has $supp K = M_{+}^{2}$ . The first stages are the actual maps

B_{f} ⋆_{1} W_{g} W_{g} ⋆_{1} B_{f^{'}} = B_{f} W_{g} + ℏ Q_{f, g}, = W_{g} B_{f^{'}} + ℏ Q^{R}_{g, f^{'}}, Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .

Their metric derivatives are compact smooth kernels, also after the required directional insertions.

Write

S_{μν} (x, x^{'}) = \partial_{μ}^{x} \partial_{ν}^{x^{'}} Δ_{m}^{+} (x - x^{'})

for the fixed scalar contraction kernel, and

T_{f, f^{'}} = ⟨ f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}), w^{μ} (x) w^{ν} (x^{'}) ⟩ .

The three one-edge outputs

G_{12} = Q_{f, g} B_{f^{'}}, G_{23} = B_{f} Q^{R}_{g, f^{'}}, G_{13} = T_{f, f^{'}} W_{g}

exist and belong to $P_{Σ}$ .

For $V = f \otimes g \otimes f^{'}$ , the three two-edge values are the already defined and admitted functionals

G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .

Their defining wavefront products, compact coefficient kernels, admission decompositions, and all required descendants have already been established in IX-Free. Their fixed-support test-family continuity is inherited.

The maps $q_{L}, q_{R}$ defined below are injective. The inherited degree-two second-stage maps are

L_{1} (Q_{f, g}, B_{f^{'}}) R_{1} (B_{f}, Q^{R}_{g, f^{'}}) = G_{12, 23} + G_{12, 13}, = G_{12, 23} + G_{13, 23}, L_{2} (Q_{f, g}, B_{f^{'}}) R_{2} (B_{f}, Q^{R}_{g, f^{'}}) = N_{c} (f, g, f^{'}) 1, = N_{c} (f, g, f^{'}) 1.

They are maps on actual arguments, with inherited reversal and presentation independence.

Displayed formulas in this passage

IX-Free-2@1#eq:firstL · line 414
$B_{f} ⋆_{1} W_{g} W_{g} ⋆_{1} B_{f^{'}} = B_{f} W_{g} + ℏ Q_{f, g}, = W_{g} B_{f^{'}} + ℏ Q^{R}_{g, f^{'}}, Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .$
IX-Free-2@1#eq:firstR · line 414
$B_{f} ⋆_{1} W_{g} W_{g} ⋆_{1} B_{f^{'}} = B_{f} W_{g} + ℏ Q_{f, g}, = W_{g} B_{f^{'}} + ℏ Q^{R}_{g, f^{'}}, Q_{f, g} Q^{R}_{g, f^{'}} = - 2 ⟨ f (x) g (y) K (x, y), \partial_{t} ϕ (x) h_{00} (y) ⟩, = - 2 ⟨ g (y) f^{'} (x^{'}) K (x^{'}, y), h_{00} (y) \partial_{t^{'}} ϕ (x^{'}) ⟩ .$
IX-Free-2@1#eq:oneedges · line 443
$G_{12} = Q_{f, g} B_{f^{'}}, G_{23} = B_{f} Q^{R}_{g, f^{'}}, G_{13} = T_{f, f^{'}} W_{g}$
IX-Free-2@1#eq:G1223 · line 453
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-Free-2@1#eq:G1213 · line 453
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-Free-2@1#eq:G1323 · line 453
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-Free-2@1#eq:degree2L · line 474
$L_{1} (Q_{f, g}, B_{f^{'}}) R_{1} (B_{f}, Q^{R}_{g, f^{'}}) = G_{12, 23} + G_{12, 13}, = G_{12, 23} + G_{13, 23}, L_{2} (Q_{f, g}, B_{f^{'}}) R_{2} (B_{f}, Q^{R}_{g, f^{'}}) = N_{c} (f, g, f^{'}) 1, = N_{c} (f, g, f^{'}) 1.$
IX-Free-2@1#eq:degree2R · line 474
$L_{1} (Q_{f, g}, B_{f^{'}}) R_{1} (B_{f}, Q^{R}_{g, f^{'}}) = G_{12, 23} + G_{12, 13}, = G_{12, 23} + G_{13, 23}, L_{2} (Q_{f, g}, B_{f^{'}}) R_{2} (B_{f}, Q^{R}_{g, f^{'}}) = N_{c} (f, g, f^{'}) 1, = N_{c} (f, g, f^{'}) 1.$
unlabeled · line 432
$S_{μν} (x, x^{'}) = \partial_{μ}^{x} \partial_{ν}^{x^{'}} Δ_{m}^{+} (x - x^{'})$
unlabeled · line 437
$T_{f, f^{'}} = ⟨ f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}), w^{μ} (x) w^{ν} (x^{'}) ⟩ .$

Assumption: Locked stages and inherited graph outputs

Displayed formulas in this passage

Nearby in this section