Manuscript X-C@3 · X-C-v3.tex · section · 2260f5f8

The completed Yang–Feldman test

C4: completed test, three-term variation, and cancellation › The completed Yang–Feldman test · lines 967–1003

X-C@3#eq:completed-test, X-C@3#eq:completed-field

For $c [q] = f + κ c_{1} + κ^{2} c_{2}$ , set

e [q] e_{1} e_{2} = R_{q} c [q] = f + κ e_{1} + κ^{2} e_{2}, = c_{1} - L_{1, g} E_{m}^{A} f, = c_{2} - L_{1, g} E_{m}^{A} e_{1} - L_{2, g} E_{m}^{A} f .

The last line is the coefficient recursion obtained from $(1 + L [q] E_{m}^{A}) e = c$ . Expanding it gives

e_{2} = c_{2} - L_{1, g} E_{m}^{A} c_{1} - L_{2, g} E_{m}^{A} f + L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} f .

Consequently

A [q] = ϕ (e [q]) = ϕ (f) + κ A_{1} + κ^{2} A_{2}

has precisely the four second-order terms in the release box.

Every $e_{j}$ is a smooth compact test, with

supp e_{j} \subset K \cup (supp g \cap J^{-} (K)) .

The background dependence is continuous at fixed $ε$ . By transposition and the copied bare expansion, $⟨ e_{2}, u ⟩$ is X-A's complete classical second coefficient for every smooth free input $u$ . This observation is a normalization check, not yet the Ward proof.

Displayed formulas in this passage

X-C@3#eq:completed-test · line 970
$e [q] e_{1} e_{2} = R_{q} c [q] = f + κ e_{1} + κ^{2} e_{2}, = c_{1} - L_{1, g} E_{m}^{A} f, = c_{2} - L_{1, g} E_{m}^{A} e_{1} - L_{2, g} E_{m}^{A} f .$
X-C@3#eq:completed-field · line 985
$A [q] = ϕ (e [q]) = ϕ (f) + κ A_{1} + κ^{2} A_{2}$
unlabeled · line 981
$e_{2} = c_{2} - L_{1, g} E_{m}^{A} c_{1} - L_{2, g} E_{m}^{A} f + L_{1, g} E_{m}^{A} L_{1, g} E_{m}^{A} f .$
unlabeled · line 993
$supp e_{j} \subset K \cup (supp g \cap J^{-} (K)) .$