Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

A unique comparison, rather than false equality

Causal support, switch equality, and classical comparison › A unique comparison, rather than false equality · lines 1348–1367

VII-A-ST@1#eq:formalGreen

Fix a metric jet and two switches. Write

D_{i} = P_{m} + κ V_{i} + κ^{2} W_{i} .

Let $Sol_{i}$ be its formal scalar solution space through order two, with arbitrary smooth formal Cauchy data. Define

G_{i} = E_{m} - κ E_{m} V_{i} E_{m} + κ^{2} (E_{m} V_{i} E_{m} V_{i} E_{m} - E_{m} W_{i} E_{m}) (mod κ^{3}) .

Multiplication gives $D_{i} G_{i} = id (mod κ^{3})$ on smooth sources. Every coefficient output has zero Cauchy data.

Displayed formulas in this passage

VII-A-ST@1#eq:formalGreen · line 1357
$G_{i} = E_{m} - κ E_{m} V_{i} E_{m} + κ^{2} (E_{m} V_{i} E_{m} V_{i} E_{m} - E_{m} W_{i} E_{m}) (mod κ^{3}) .$
unlabeled · line 1351
$D_{i} = P_{m} + κ V_{i} + κ^{2} W_{i} .$