Manuscript Em02-A@6 · Em02-A-v6.tex · section · cb2f0a9a

Route B: geometry, support zeros, and the contribution ledger

Route B: geometry, support zeros, and the contribution ledger · lines 502–552

Em02-A@6#eq:chart, Em02-A@6#eq:cones, Em02-A@6#eq:internal-cones, Em02-A@6#eq:density

Choose a local sphere frame

ℓ = (1, n), k = (1, - n), E_{A} = (0, e_{A}), A = 1, 2,

with $ℓ \cdot k = 2$ and $E_{A} \cdot E_{B} = - δ_{A B}$ . Use

z = a + r, b = a + β ℓ + u k, y = a + α ℓ + v k + ξ^{A} E_{A}, r = γ ℓ + υ k + ω^{A} E_{A} .

Fix a compact regular chart with positive lower bounds for $α$ , $β - α$ , $β$ , and $β - u$ . Where a non-tip formula requires $β - γ > 0$ , that restriction is imposed separately.

Put

q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2}, q_{z} = (b - z)^{2} .

Then

q_{R} r^{2} = 4 β u, = 4 γ υ - ∣ ω ∣^{2}, q_{1} q_{z} = 4 α v - ∣ ξ ∣^{2}, = 4 (β - γ) (u - υ) - ∣ ω ∣^{2} . q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},

The density is

d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d τ d^{4} r d u d v d^{2} ξ, d τ = d^{4} a d Ω d α d β,

and $d^{4} r = 2 d γ d υ d^{2} ω$ . The inverse chart and density verification are given in Appendix app:geometry.

Displayed formulas in this passage

Em02-A@6#eq:chart · line 512
$z = a + r, b = a + β ℓ + u k, y = a + α ℓ + v k + ξ^{A} E_{A}, r = γ ℓ + υ k + ω^{A} E_{A} .$
Em02-A@6#eq:cones · line 531
$q_{R} r^{2} = 4 β u, = 4 γ υ - ∣ ω ∣^{2}, q_{1} q_{z} = 4 α v - ∣ ξ ∣^{2}, = 4 (β - γ) (u - υ) - ∣ ω ∣^{2} . q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},$
Em02-A@6#eq:internal-cones · line 531
$q_{R} r^{2} = 4 β u, = 4 γ υ - ∣ ω ∣^{2}, q_{1} q_{z} = 4 α v - ∣ ξ ∣^{2}, = 4 (β - γ) (u - υ) - ∣ ω ∣^{2} . q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},$
Em02-A@6#eq:density · line 541
$d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d τ d^{4} r d u d v d^{2} ξ, d τ = d^{4} a d Ω d α d β,$
unlabeled · line 506
$ℓ = (1, n), k = (1, - n), E_{A} = (0, e_{A}), A = 1, 2,$
unlabeled · line 526
$q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2}, q_{z} = (b - z)^{2} .$