Manuscript IX-D@3 · IX-D-v3.tex · section · 3ad4586e

Available word coefficients

D0: the constructed IX-Free-2 sector › Available word coefficients · lines 668–737

IX-D@3#eq:oneedges, IX-D@3#eq:G1223, IX-D@3#eq:G1213, IX-D@3#eq:G1323, IX-D@3#eq:word

The inherited first stages are

B_{f} ⋆_{c} W_{g} = B_{f} W_{g} + ℏ Q_{f, g}, W_{g} ⋆_{c} B_{f^{'}} = W_{g} B_{f^{'}} + ℏ Q^{R}_{g, f^{'}} .

Write

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

and

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

The one-edge outputs are

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

For $V = f \otimes g \otimes f^{'}$ , the two-edge outputs are

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 distributional existence, compact kernels, admission, and required descendants are inherited.

The common constructed word is

A_{c} (f, g, f^{'}) = (B_{f} ⋆_{c} W_{g}) ⋆_{c} B_{f^{'}} = B_{f} ⋆_{c} (W_{g} ⋆_{c} B_{f^{'}}) = B_{f} W_{g} B_{f^{'}} + ℏ (G_{12} + G_{13} + G_{23}) + ℏ^{2} (G_{12, 23} + G_{12, 13} + G_{13, 23}) + ℏ^{3} N_{c} (f, g, f^{'}) 1.

This is an equality of existing partial-product associations, not an assertion about unspecified triples.

The sign ledger remains the inherited one: each partial has factor $- 2$ , the complete word has two insertion signs with product $+ 1$ , and the middle-slot assignment factor is $2$ , not $4$ .

Displayed formulas in this passage

IX-D@3#eq:oneedges · line 689
$G_{12} = Q_{f, g} B_{f^{'}}, G_{23} = B_{f} Q^{R}_{g, f^{'}}, G_{13} = T_{f, f^{'}} W_{g} .$
IX-D@3#eq:G1223 · line 696
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-D@3#eq:G1213 · line 696
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-D@3#eq:G1323 · line 696
$G_{12, 23} G_{12, 13} G_{13, 23} = 2 ⟨ V K (x, y) K (x^{'}, y), \partial_{t} ϕ (x) \partial_{t^{'}} ϕ (x^{'}) ⟩, = 2 ⟨ V K (x, y) S_{0 ν} (x, x^{'}), h_{00} (y) w^{ν} (x^{'}) ⟩, = 2 ⟨ V S_{μ 0} (x, x^{'}) K (x^{'}, y), w^{μ} (x) h_{00} (y) ⟩ .$
IX-D@3#eq:word · line 714
$A_{c} (f, g, f^{'}) = (B_{f} ⋆_{c} W_{g}) ⋆_{c} B_{f^{'}} = B_{f} ⋆_{c} (W_{g} ⋆_{c} B_{f^{'}}) = B_{f} W_{g} B_{f^{'}} + ℏ (G_{12} + G_{13} + G_{23}) + ℏ^{2} (G_{12, 23} + G_{12, 13} + G_{13, 23}) + ℏ^{3} N_{c} (f, g, f^{'}) 1.$
unlabeled · line 671
$B_{f} ⋆_{c} W_{g} = B_{f} W_{g} + ℏ Q_{f, g}, W_{g} ⋆_{c} B_{f^{'}} = W_{g} B_{f^{'}} + ℏ Q^{R}_{g, f^{'}} .$
unlabeled · line 678
$S_{μν} (x, x^{'}) = \partial_{μ}^{x} \partial_{ν}^{x^{'}} Δ_{m}^{+} (x - x^{'})$
unlabeled · line 683
$T_{f, f^{'}} = ⟨ f (x) f^{'} (x^{'}) S_{μν} (x, x^{'}), w^{μ} (x) w^{ν} (x^{'}) ⟩ .$