Manuscript Em01-G@5 · Em01-G-v5.tex · theorem · 3fbfe951

Theorem: Evaluated supported inverse-cubic coefficient

Supported assembly and its leading coefficient · lines 1362–1395 · proof included or linked

Em01-G@5#thm:supported-leading, Em01-G@5#eq:supported-leading

For $w_{0} = (a, a + β ℓ, a, a + α ℓ)$ ,

A_{- 3}^{supp} (Φ) = - \frac{1}{512 π ^{3}} \int M_{3} (n) f (a + β ℓ) J \sum P_{J} (n) (ℓ^{ν} \partial_{ν} g) (a + α ℓ) Φ_{J} (w_{0}) d ϑ .

Proof. Only $j = b = 2$ kinetic terms have $q = 3$ . Three profile integrations by parts give

\int ν_{2, μ}^{K} lo g (B_{1} - B_{2} + i 0) d H = - ℓ_{μ} M_{3} .

Equations (eq:v0), (eq:projection0), and (eq:projection1) leave

\frac{1}{512 π ^{3}} \int M_{3} f (a + β ℓ) J \sum P_{J} g (a + α ℓ) \partial_{α} [Φ_{J} (w_{0})] d ϑ .

Compact tangential integration by parts in $α$ yields (eq:supported-leading).

Displayed formulas in this passage

Em01-G@5#eq:supported-leading · line 1365
$A_{- 3}^{supp} (Φ) = - \frac{1}{512 π ^{3}} \int M_{3} (n) f (a + β ℓ) J \sum P_{J} (n) (ℓ^{ν} \partial_{ν} g) (a + α ℓ) Φ_{J} (w_{0}) d ϑ .$
unlabeled · line 1382
$\int ν_{2, μ}^{K} lo g (B_{1} - B_{2} + i 0) d H = - ℓ_{μ} M_{3} .$
unlabeled · line 1388
$\frac{1}{512 π ^{3}} \int M_{3} f (a + β ℓ) J \sum P_{J} g (a + α ℓ) \partial_{α} [Φ_{J} (w_{0})] d ϑ .$