Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Affine checks of the third-derivative coefficient

Integrated-vertex jets and the known six rows › Affine checks of the third-derivative coefficient · lines 1395–1431

These checks are local: weights may be cut off outside a fixed neighborhood of all relevant convolution fibers.

For $ϕ = 1$ ,

P_{3} = \frac{κ}{16 π} .

This agrees with $M_{δ} = J_{δ} (S; 1) /2$ .

For $ϕ = x^{0}$ , symmetry gives the exact identity

J_{δ} (S; x^{0}) = \frac{S ^{0}}{2} J_{δ} (S; 1),

hence

M_{δ} = - \frac{1}{2} D J_{δ} (S; 1) + \frac{S ^{0}}{4} J_{δ} (S; 1) .

Formula (eq:P3) gives

P_{3} = \frac{κ}{32 π} (q - 3/ q),

which agrees with direct differentiation.

For $ϕ = x^{j}$ ,

J_{δ} (S; x^{j}) = \frac{S ^{j}}{2} J_{δ} (S; 1),

and (eq:P3) gives

P_{3} = \frac{κ n _{j}}{32 π} (q + 3/ q) .

These identities audit signs and factors; the arbitrary-test proof is Proposition prop:vertex-jets.

Displayed formulas in this passage

unlabeled · line 1401
$P_{3} = \frac{κ}{16 π} .$
unlabeled · line 1407
$J_{δ} (S; x^{0}) = \frac{S ^{0}}{2} J_{δ} (S; 1),$
unlabeled · line 1411
$M_{δ} = - \frac{1}{2} D J_{δ} (S; 1) + \frac{S ^{0}}{4} J_{δ} (S; 1) .$
unlabeled · line 1416
$P_{3} = \frac{κ}{32 π} (q - 3/ q),$
unlabeled · line 1422
$J_{δ} (S; x^{j}) = \frac{S ^{j}}{2} J_{δ} (S; 1),$
unlabeled · line 1426
$P_{3} = \frac{κ n _{j}}{32 π} (q + 3/ q) .$