Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

Fixed width and supports

Frozen data and the inherited scalar completion › Fixed width and supports · lines 290–336

X-E-ST@1#eq:separation

Fix

m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .

Let $U$ be a nonempty relatively compact open set with

U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .

The formal group $G_{p}$ is generated through second order by $κ ξ + κ^{2} ζ$ , with $ξ, ζ \in g_{p}$ . Set

K = supp f, S = supp g \cap J_{η}^{-} (K), C = K \cup S, D_{f} = J_{η}^{-} (K) \cap M_{T_{*}} .

Choose $δ > 0$ so that

K \cup supp g \cup U \subset {2 δ < t < T_{*}} .

Fix $0 < ε < δ$ , once and for all, and use

σ (t) = χ (t / ε),

where $χ$ is the smooth predecessor profile, zero on $(- \infty, 0]$ , one on $[1, \infty)$ , and flat at both endpoints. No width-removal theorem is sought.

The backgrounds $h, k$ are fixed smooth real symmetric tensors on the completion. No equations or spatial decay are imposed on them. Write

q = η + κh + κ^{2} k .

Formal identities for observables are understood modulo $κ^{3}$ . Variations of the exact action are derived before truncation; negative powers of $κ$ in its normalization are not manipulated in a nilpotent ring.

Displayed formulas in this passage

X-E-ST@1#eq:separation · line 313
$K \cup supp g \cup U \subset {2 δ < t < T_{*}} .$
unlabeled · line 293
$m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .$
unlabeled · line 299
$U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 307
$K = supp f, S = supp g \cap J_{η}^{-} (K), C = K \cup S, D_{f} = J_{η}^{-} (K) \cap M_{T_{*}} .$
unlabeled · line 319
$σ (t) = χ (t / ε),$
unlabeled · line 329
$q = η + κh + κ^{2} k .$