Manuscript X-D@3 · X-D-v3.tex · section · b7774529

D1: topology sheet and separation from the initial surface

D1: topology sheet and separation from the initial surface · lines 554–619

X-D@3#eq:seminorms

The compact $C$ above is independent of $ε$ and lies above $t = 2 δ$ . Define

D_{C} (M) = {d \in C_{c}^{\infty} (M; C) : supp d \subset C},

with seminorms

p_{N} (d) = ∣ α ∣ \leq N max x \in M sup ∣ \partial^{α} d (x) ∣.

These are ambient smooth derivatives. No topology on a nonsmooth causal boundary is silently substituted.

The linear output topology is the quotient topology of

Q_{m} = D (M) / P_{m} D (M) .

The inclusion $D_{C} (M) \to D (M)$ and the quotient map are continuous. An individual representative seminorm need not descend to the quotient; convergence passes through the continuous quotient map.

The equation subspace is closed:

P_{m} D (M) = ker (E_{m}^{A} - E_{m}^{R}),

by the Green-operator exact sequence BGP (Theorem 3.4.7). Hence $Q_{m}$ is Hausdorff. No additional topology on the full Wick algebra is required for the linear coefficients considered here.

{}1.12

Item	Declaration
Inputs	Fixed $f, g, h, k$ and fixed $U$ , with (eq:separation)
Object	The existing completed tests $d_{1, ε}, d_{2, ε}$
Topology	$D_{C} (M)$ , then the original quotient $Q_{m}$
Support	One compact $C$ , independent of $ε$
Stop clause	Observation or interaction supports meeting $Σ$ define a different family, not covered here
Candidate	Named after the limiting coordinate data are computed
Not asserted	Smooth coordinate convergence through $t = 0$ ; joint sharp admission
Not opened	New $R_{n}$ , quantum metric, triangle subtraction, convolution repair

There will generally be no smooth convergence of coordinate velocities on the entire closed slab: the ramp velocities at $t = 0$ are zero, while their limiting right-hand values need not be. This is compatible with convergence of the completed tests, whose supports remain above the surface.

Displayed formulas in this passage

X-D@3#eq:seminorms · line 564
$p_{N} (d) = ∣ α ∣ \leq N max x \in M sup ∣ \partial^{α} d (x) ∣.$
unlabeled · line 559
$D_{C} (M) = {d \in C_{c}^{\infty} (M; C) : supp d \subset C},$
unlabeled · line 573
$Q_{m} = D (M) / P_{m} D (M) .$
unlabeled · line 582
$P_{m} D (M) = ker (E_{m}^{A} - E_{m}^{R}),$