Manuscript Em01-E@2 · Em01-E-v2.tex · corollary · 54f8f8c2

Corollary: Cancellation of the inverse-scale u-jet

Evaluated complete degenerate specialization › An additional complete-coefficient cancellation · lines 2071–2107 · proof included or linked

Em01-E@2#cor:appcancel, Em01-E@2#eq:appAsimple

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Write $φ (τ; u, v, X) = F_{0} (τ; u, v, X)$ . Then

A (Φ) = - \frac{i Q _{p}}{2} \int d τ [- \frac{1}{4} \partial_{v} φ + \frac{B}{2} Δ_{X} φ + 2 (2 a^{0} + A + B) φ]_{0} .

The normal derivative $\partial_{u} φ$ cancels in the complete inverse-scale coefficient. This cancellation does not remove its nonzero zeroth-order term.

Proof. The fixed-frame identity gives

F_{1} = - \frac{1}{2} (\partial_{B} + \partial_{u}) φ + 2 (b^{0} + y^{0}) φ .

Thus $\frac{1}{2} \partial_{u} F_{0} + F_{1}$ equals

- \frac{1}{2} \partial_{B} φ + 2 (b^{0} + y^{0}) φ .

The first term integrates to zero in $B$ , and on the attachment $b^{0} + y^{0} = 2 a^{0} + A + B$ .

Displayed formulas in this passage

Em01-E@2#eq:appAsimple · line 2075
$A (Φ) = - \frac{i Q _{p}}{2} \int d τ [- \frac{1}{4} \partial_{v} φ + \frac{B}{2} Δ_{X} φ + 2 (2 a^{0} + A + B) φ]_{0} .$
unlabeled · line 2095
$F_{1} = - \frac{1}{2} (\partial_{B} + \partial_{u}) φ + 2 (b^{0} + y^{0}) φ .$
unlabeled · line 2102
$- \frac{1}{2} \partial_{B} φ + 2 (b^{0} + y^{0}) φ .$