Manuscript XIII-C@1 · XIII-C.tex · lemma · e6556893

Lemma: Eleven-branch decomposition

C1: complete incidence inventory for the selected pair · lines 534–560 · proof included or linked

XIII-C@1#lem:branches, XIII-C@1#eq:branches

The displayed lift is exactly $O = \sum_{j = 1}^{11} B_{j}$ , where

B_{1} B_{3} B_{5} B_{7} B_{9} B_{11} = - \int_{f} E_{m}^{R} (V_{1, g} [γ] p), = - \int_{f} X^{μ} \partial_{μ} p, = \int_{f} E_{0}^{R} (U_{σ}_{ε} [γ] X^{μ}) \partial_{μ} ϕ, = \frac{1}{2} \int_{f} X^{μ} X^{ν} \partial_{μ} \partial_{ν} ϕ, = \int_{f} p_{1} [r_{D}; ϕ], = \int_{f} p_{1} [r_{g}; ϕ] . B_{2} B_{4} B_{6} B_{8} B_{10} = - \int_{f} E_{m}^{R} (V_{2, g} [γ, 0] ϕ), = \int_{f} X^{ν} \partial_{ν} X^{μ} \partial_{μ} ϕ, = \int_{f} E_{0}^{R} (\partial_{α} (σ_{ε} a_{2}^{α μ} [γ])) \partial_{μ} ϕ, = - \int_{f} X_{1}^{μ} [r_{D}] \partial_{μ} ϕ, = - \int_{f} X_{1}^{μ} [r_{g}] \partial_{μ} ϕ,

Proof. Expand the two parts of $p_{2} [γ, 0; ϕ]$ , the two parts of $- X_{2} [γ, 0] \cdot \partial ϕ$ , and the other three terms of the second completion. The affine second-slot identity supplies $B_{8} + B_{9} = A_{1} (r_{D}; ϕ) = - C_{3, D}$ . The two legs of the scalar-sourced slot give $B_{10} + B_{11}$ .

Displayed formulas in this passage

XIII-C@1#eq:branches · line 537
$B_{1} B_{3} B_{5} B_{7} B_{9} B_{11} = - \int_{f} E_{m}^{R} (V_{1, g} [γ] p), = - \int_{f} X^{μ} \partial_{μ} p, = \int_{f} E_{0}^{R} (U_{σ}_{ε} [γ] X^{μ}) \partial_{μ} ϕ, = \frac{1}{2} \int_{f} X^{μ} X^{ν} \partial_{μ} \partial_{ν} ϕ, = \int_{f} p_{1} [r_{D}; ϕ], = \int_{f} p_{1} [r_{g}; ϕ] . B_{2} B_{4} B_{6} B_{8} B_{10} = - \int_{f} E_{m}^{R} (V_{2, g} [γ, 0] ϕ), = \int_{f} X^{ν} \partial_{ν} X^{μ} \partial_{μ} ϕ, = \int_{f} E_{0}^{R} (\partial_{α} (σ_{ε} a_{2}^{α μ} [γ])) \partial_{μ} ϕ, = - \int_{f} X_{1}^{μ} [r_{D}] \partial_{μ} ϕ, = - \int_{f} X_{1}^{μ} [r_{g}] \partial_{μ} ϕ,$

Lemma: Eleven-branch decomposition

Displayed formulas in this passage

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