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

Conventions

Setting, explicit interfaces, and normalization › Conventions · lines 234–282

Work on ambient Minkowski space $M = R^{4}$ , with

η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.

All indices are moved with $η$ , and parentheses on indices include the factor $1/2$ . The symmetric-tensor fiber is

V = Sym^{2} (R^{4})^{*}, b (A, C) = A_{μν} C^{μν} .

All tensor and distributional pairings are bilinear, with no complex conjugation.

Trace reversal and the free tensor operator are

P A = A - \frac{1}{2} η tr_{η} A, P^{2} = id, K = - □ P .

For multi-indices $p, q, γ \in N^{4}$ ,

\partial^{p} = \partial_{0}^{p_{0}} \partial_{1}^{p_{1}} \partial_{2}^{p_{2}} \partial_{3}^{p_{3}}, ∣ p ∣ = α = 0 \sum^{3} p_{α} .

The zero multi-index is denoted by $0$ , and $e_{α}$ is the $α$ -th unit multi-index.

The notation is deliberately separated:

S (z) is the scalar edge distribution, Σ_{α β} (y) = \partial_{α} \partial_{β} σ_{ε} (y) is the preparation jet .

A positive dilation parameter is written $a > 0$ . It is not a witness frequency. The scalar configuration is $φ$ , and a compact distributional test is usually $ψ$ .

Displayed formulas in this passage

unlabeled · line 237
$η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ.$
unlabeled · line 244
$V = Sym^{2} (R^{4})^{*}, b (A, C) = A_{μν} C^{μν} .$
unlabeled · line 253
$P A = A - \frac{1}{2} η tr_{η} A, P^{2} = id, K = - □ P .$
unlabeled · line 262
$\partial^{p} = \partial_{0}^{p_{0}} \partial_{1}^{p_{1}} \partial_{2}^{p_{2}} \partial_{3}^{p_{3}}, ∣ p ∣ = α = 0 \sum^{3} p_{α} .$
unlabeled · line 272
$S (z) is the scalar edge distribution, Σ_{α β} (y) = \partial_{α} \partial_{β} σ_{ε} (y) is the preparation jet .$