Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

The full intersection pairing

Intersection and sector audit › The full intersection pairing · lines 1636–1672

VII-D@1#eq:intersectionpairing

For a test supported in an incidence-coordinate patch, the exact pairing corresponding to (eq:Adef) is

⟨ A (r), Φ ⟩ = c_{0} r^{3} ω (r) Os \int \frac{( a + σ ) ^{2} Φ e ^{i Θ_{r}}}{P ^{2} P ^{'2}} \times d^{3} z d Ω (n) d t d s d d d σ d^{3} w d Ω (ν) \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}},

where $t^{'} = t - d > 0$ and $Θ_{r}$ is (eq:intersectionphase). The oscillatory notation means the Cartesian-Fourier distribution (eq:Adef) rewritten by the cutoff-and-coordinate procedure in Lemma A. It is not a new product of singular distributions.

The leading Fourier inversion imposes

x - y = a ν, x^{'} - y = b ν .

Only after that inversion may one conclude

ν = n, σ = 0, w = 0.

The positive radius $a = t + s$ , not $a - b = d$ , determines the first angular constraint. Neither this identification nor its density requires division by $d$ .

Displayed formulas in this passage

VII-D@1#eq:intersectionpairing · line 1640
$⟨ A (r), Φ ⟩ = c_{0} r^{3} ω (r) Os \int \frac{( a + σ ) ^{2} Φ e ^{i Θ_{r}}}{P ^{2} P ^{'2}} \times d^{3} z d Ω (n) d t d s d d d σ d^{3} w d Ω (ν) \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}},$
unlabeled · line 1661
$x - y = a ν, x^{'} - y = b ν .$
unlabeled · line 1666
$ν = n, σ = 0, w = 0.$