Manuscript XIX-D@1 · XIX-D.tex · lemma · 1f47e33b

Lemma: Formal-order persistence

Retarded memory, formal order, and gauge closure › Positive formal quantum order cannot remove a fixed classical coefficient · lines 883–899 · proof included or linked

XIX-D@1#lem:formal

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Let $E$ be a complex vector space and $D_{0} \neq = 0$ . For every $D_{+} \in ℏ E [[ℏ]]$ ,

D_{0} + D_{+} \neq = 0 in E [[ℏ]] .

The same conclusion holds at a fixed coupling coefficient $κ^{r}$ in a doubly formal series.

Proof. The coefficient of $ℏ^{0}$ is $D_{0}$ , unchanged by $D_{+}$ . For the doubly formal statement, extract the coefficient of $κ^{r} ℏ^{0}$ .

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$D_{0} + D_{+} \neq = 0 in E [[ℏ]] .$

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section: Positive formal quantum order cannot remove a fixed classical coefficient