Manuscript XVI-K@1 · XVI-K.tex · section · aa7553bb

Constant-slot and background regressions

Computational artifacts and the scope of their verification › Constant-slot and background regressions · lines 2092–2189

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For $h_{*} = (w^{1})^{2} η$ , the supplied nonzero independent constant-slot entries are

L_{00} = - 1, L_{04} = L_{07} = L_{09} = - 1, L_{11} = 12, L_{22} = L_{33} = 4, L_{44} = - 9, L_{47} = L_{49} = 1, L_{55} = L_{66} = - 12, L_{77} = L_{99} = - 1, L_{79} = 1, L_{88} = - 4.

Every unlisted entry is zero, and the matrix is symmetric. Thus the reported mixed value is

D^{2} ℓ_{3} (h_{*}) [η, B_{4}] = L_{04} - L_{44} - L_{74} - L_{94} = 6.

This value is at the Hessian level. It is not the scalar coefficient of the decorated graph.

The supplied constant-slot decorated output, translated from its S00, S01, \ notation to $Σ$ , is

K_{00}^{00} K_{00}^{01} K_{00}^{11} K_{00}^{12} K_{00}^{22} K_{00}^{23} K_{00}^{33} = 14 Σ_{11} + 10 Σ_{22} + 10 Σ_{33}, = - 14 Σ_{01}, = 14 (Σ_{00} - Σ_{22} - Σ_{33}), = 14 Σ_{12}, = 10 Σ_{00} - 14 Σ_{11} - 10 Σ_{33}, = 10 Σ_{23}, = 10 Σ_{00} - 14 Σ_{11} - 10 Σ_{22} . K_{00}^{02} K_{00}^{13} = - 10 Σ_{02}, = 14 Σ_{13}, K_{00}^{03} = - 10 Σ_{03},

Selected differentiated-slot entries in the supplied general coefficient list are

L^{0, e_{0}}_{(00) (00)} L^{0, 2 e_{1}}_{(00) (00)} L^{e_{1}, e_{1}}_{(00) (00)} L^{e_{0}, e_{0}}_{(33) (33)} L^{2 e_{0}, 0}_{(33) (33)} = 3 h_{01, 1} + 3 h_{02, 2} + 3 h_{03, 3} - \frac{3}{2} (h_{11, 0} + h_{22, 0} + h_{33, 0}), = - \frac{3}{2} h_{00} + \frac{1}{2} h_{11} - \frac{1}{2} h_{22} - \frac{1}{2} h_{33}, = - 3 h_{00} + h_{11} - h_{22} - h_{33}, = h_{00} + h_{11} + h_{22} - 3 h_{33}, = \frac{1}{2} (h_{00} + h_{11} + h_{22} - 3 h_{33}) .

Here $h_{ab, c} = \partial_{c} h_{ab}$ . These entries illustrate the derivative grading; the complete list remains in the unchanged supplied output.

For $s = w^{1}$ , selected supplied decorated background entries are

K^{00}_{0, e_{1}} K^{11}_{0, e_{1}} K^{00}_{e_{1}, e_{1}} K^{11}_{0, 2 e_{0}} = 2 s (8 Σ_{11} + 13 Σ_{22} + 13 Σ_{33}), = 16 s (Σ_{00} - Σ_{22} - Σ_{33}), = s^{2} (Σ_{11} + 8 Σ_{22} + 8 Σ_{33}), = - \frac{1}{2} s^{2} (Σ_{00} - 11 Σ_{22} - 11 Σ_{33}) .

They have residual degrees $s$ and $s^{2}$ , as predicted by Proposition prop:hess.

For the actual time-only source jet, the last two relevant spatial components in this selection give

K^{11}_{0, e_{1}} = 16 w^{1} σ_{ε}^{''}, K^{11}_{0, 2 e_{0}} = - \frac{1}{2} (w^{1})^{2} σ_{ε}^{''} .

These are background coefficient statements, not evaluated switched or smeared graph values.

Displayed formulas in this passage

unlabeled · line 2119
$K_{00}^{00} K_{00}^{01} K_{00}^{11} K_{00}^{12} K_{00}^{22} K_{00}^{23} K_{00}^{33} = 14 Σ_{11} + 10 Σ_{22} + 10 Σ_{33}, = - 14 Σ_{01}, = 14 (Σ_{00} - Σ_{22} - Σ_{33}), = 14 Σ_{12}, = 10 Σ_{00} - 14 Σ_{11} - 10 Σ_{33}, = 10 Σ_{23}, = 10 Σ_{00} - 14 Σ_{11} - 10 Σ_{22} . K_{00}^{02} K_{00}^{13} = - 10 Σ_{02}, = 14 Σ_{13}, K_{00}^{03} = - 10 Σ_{03},$
unlabeled · line 2144
$L^{0, e_{0}}_{(00) (00)} L^{0, 2 e_{1}}_{(00) (00)} L^{e_{1}, e_{1}}_{(00) (00)} L^{e_{0}, e_{0}}_{(33) (33)} L^{2 e_{0}, 0}_{(33) (33)} = 3 h_{01, 1} + 3 h_{02, 2} + 3 h_{03, 3} - \frac{3}{2} (h_{11, 0} + h_{22, 0} + h_{33, 0}), = - \frac{3}{2} h_{00} + \frac{1}{2} h_{11} - \frac{1}{2} h_{22} - \frac{1}{2} h_{33}, = - 3 h_{00} + h_{11} - h_{22} - h_{33}, = h_{00} + h_{11} + h_{22} - 3 h_{33}, = \frac{1}{2} (h_{00} + h_{11} + h_{22} - 3 h_{33}) .$
unlabeled · line 2165
$K^{00}_{0, e_{1}} K^{11}_{0, e_{1}} K^{00}_{e_{1}, e_{1}} K^{11}_{0, 2 e_{0}} = 2 s (8 Σ_{11} + 13 Σ_{22} + 13 Σ_{33}), = 16 s (Σ_{00} - Σ_{22} - Σ_{33}), = s^{2} (Σ_{11} + 8 Σ_{22} + 8 Σ_{33}), = - \frac{1}{2} s^{2} (Σ_{00} - 11 Σ_{22} - 11 Σ_{33}) .$
unlabeled · line 2096
$L_{00} = - 1, L_{04} = L_{07} = L_{09} = - 1, L_{11} = 12, L_{22} = L_{33} = 4, L_{44} = - 9, L_{47} = L_{49} = 1, L_{55} = L_{66} = - 12, L_{77} = L_{99} = - 1, L_{79} = 1, L_{88} = - 4.$
unlabeled · line 2109
$D^{2} ℓ_{3} (h_{*}) [η, B_{4}] = L_{04} - L_{44} - L_{74} - L_{94} = 6.$
unlabeled · line 2180
$K^{11}_{0, e_{1}} = 16 w^{1} σ_{ε}^{''}, K^{11}_{0, 2 e_{0}} = - \frac{1}{2} (w^{1})^{2} σ_{ε}^{''} .$