Manuscript IX-C-ST@1 · IX-C-ST.tex · lemma · 892f50c4

Lemma: Injective generating tests

Both first stages and the placement ledger · lines 1025–1038 · proof included or linked

IX-C-ST@1#lem:injective

The maps $f \mapsto B_{f}$ , $g \mapsto W_{g}$ are injective.

Proof. Sharp-test injectivity is included in the adapted carrier input; it also follows from the stronger local separation proved in Lemma lem:restricted below. For $W_{g}$ , polarization of $\int g v^{2} = 0$ gives $\int g uv = 0$ . Choose $v = 1$ near $supp g$ ; arbitrary compact $u$ then gives $g = 0$ .

Nearby in this section

proposition: Proposition: Reverse coefficient and its sign
section: Both first stages and the placement ledger
theorem: Theorem: Admitted first-stage products
proposition: Proposition: Fixed-support first-stage continuity
section: Both first stages and the placement ledger
proposition: Proposition: One-edge outputs
section: Both first stages and the placement ledger
proposition: Proposition: Signs and multiplicities