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

Theorem: Evaluated full positive-placement inverse-quadratic coefficient

Exact positive-placement reduction and evaluated coefficient · lines 1664–1707 · proof included or linked

Em01-G@5#thm:positive-quadratic, Em01-G@5#eq:positive-quadratic

On every fixed non-tip positive-placement localization,

A_{- 2, 5 +} (Φ) = \frac{1}{1024 π ^{4}} \int ϵ M_{2} (n) g (a + α ℓ) J \sum R_{J}^{ρ} (n) [B_{kin}^{α β κ ν} ℓ_{κ} \partial_{s^{α}} (f T_{J, β ρ ν}) + \frac{m ^{2}}{2} f ℓ^{β} T_{J, β ρ}]_{+} d μ_{+} .

Here the spacetime index $α$ inside the bracket is summed; the characteristic coordinate $α$ in the integration density is a separate scalar variable. All kinetic and mass contributions at inverse-quadratic order are included. There are no root-row contributions to this row.

Proof. By Lemma lem:positive-reduction, inverse-quadratic order comes only from (eq:positive-q2). Apply (eq:second-moment). Its negative sign cancels the outer negative sign of that numerator. The first term becomes

\frac{M _{2}}{128 π ^{2} β} g R_{J}^{ρ} B_{kin}^{α β κ ν} ℓ_{κ} \partial_{s^{α}} (f T_{J, β ρ ν}) .

The mass term has the same prefactor times $(m^{2} /2) f R_{J}^{ρ} ℓ^{β} T_{J, β ρ}$ . Multiplication by the seed density $β ϵ / (8 π^{2})$ gives (eq:positive-quadratic). All remaining exact terms have crossed order at most one and cannot contribute at inverse-quadratic order.

Displayed formulas in this passage

Em01-G@5#eq:positive-quadratic · line 1667
$A_{- 2, 5 +} (Φ) = \frac{1}{1024 π ^{4}} \int ϵ M_{2} (n) g (a + α ℓ) J \sum R_{J}^{ρ} (n) [B_{kin}^{α β κ ν} ℓ_{κ} \partial_{s^{α}} (f T_{J, β ρ ν}) + \frac{m ^{2}}{2} f ℓ^{β} T_{J, β ρ}]_{+} d μ_{+} .$
unlabeled · line 1695
$\frac{M _{2}}{128 π ^{2} β} g R_{J}^{ρ} B_{kin}^{α β κ ν} ℓ_{κ} \partial_{s^{α}} (f T_{J, β ρ ν}) .$

Theorem: Evaluated full positive-placement inverse-quadratic coefficient

Displayed formulas in this passage

Nearby in this section