Manuscript X-D-ST@1 · X-D-ST.tex · lemma · e0cf8fdb

Lemma: Reduction of the limit problem

Frozen data, completed tests, and the topology sheet › One fixed compact-test space · lines 585–626 · proof included or linked

X-D-ST@1#lem:reduction, X-D-ST@1#eq:reduction

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For tests supported in $K$ , the operators

L_{1} E_{A}, L_{2} E_{A}, L_{1} E_{A} L_{1} E_{A}

are continuous into $D_{C}$ , with the causal support specified by (eq:C).

If $c_{j, ε} \to c_{j, 0}$ in $D_{K}$ , then the corresponding completed tests converge in $D_{C}$ . More precisely,

d_{1, ε} - d_{1, 0} d_{2, ε} - d_{2, 0} = c_{1, ε} - c_{1, 0}, = c_{2, ε} - c_{2, 0} - L_{1} E_{A} (c_{1, ε} - c_{1, 0}) .

For every $N$ , there exist $M_{N}$ and $A_{N}$ such that

p_{N} (L_{1} E_{A} a) \leq A_{N} p_{M_{N}} (a), a \in D_{K} .

Proof. Advanced propagation sends a compact smooth test to a smooth function supported in its causal past. Each $L_{j}$ has compact smooth coefficients supported in $supp g$ . For the iteration, the second causal past remains in $J_{η}^{-} (K)$ . Green continuity, multiplication, and differentiation give continuous maps between the stated fixed-support Fréchet spaces. Continuity supplies some finite seminorm loss; no universal derivative-loss count is assumed. Subtraction of (eq:completed) gives (eq:reduction).

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X-D-ST@1#eq:reduction · line 597
$d_{1, ε} - d_{1, 0} d_{2, ε} - d_{2, 0} = c_{1, ε} - c_{1, 0}, = c_{2, ε} - c_{2, 0} - L_{1} E_{A} (c_{1, ε} - c_{1, 0}) .$
unlabeled · line 588
$L_{1} E_{A}, L_{2} E_{A}, L_{1} E_{A} L_{1} E_{A}$
unlabeled · line 608
$p_{N} (L_{1} E_{A} a) \leq A_{N} p_{M_{N}} (a), a \in D_{K} .$

Lemma: Reduction of the limit problem

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