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

A single partition convention

Route B: geometry, support zeros, and the contribution ledger › A single partition convention · lines 619–675

Em02-A@6#eq:partition, Em02-A@6#eq:postadjoint-partition

Choose $b_{0} \in C^{\infty} ([0, \infty); [0, 1])$ with $b_{0} = 1$ on $[0, 1]$ and $b_{0} = 0$ on $[4, \infty)$ . A fixed coincidence partition is

θ_{tip} = b_{0} (∣ r ∣_{E}^{2} / ϵ_{0}^{2}), θ_{away} = 1 - θ_{tip},

where $ϵ_{0}$ is independent of $δ$ .

For $L > 0$ , let

a_{j, δ} = b_{0} (q_{j}^{2} / (L^{2} δ^{2})), j = 1, 2, a_{R, δ} = b_{0} (q_{R}^{2} / (L^{2} δ^{2})) .

Use the exact five-sector partition

θ_{00, δ} θ_{0 R, δ} θ_{10, δ} θ_{01, δ} θ_{11, δ} = a_{1, δ} a_{2, δ} a_{R, δ}, = a_{1, δ} a_{2, δ} (1 - a_{R, δ}), = a_{1, δ} (1 - a_{2, δ}), = (1 - a_{1, δ}) a_{2, δ}, = (1 - a_{1, δ}) (1 - a_{2, δ}) .

Their sum is one.

Throughout the main calculation, these functions multiply already differentiated seed amplitudes. Thus

v_{e, j, δ} (Φ) = - i I, J \sum I_{e} (ϑ_{e, j, δ} P_{e}^{I, J} [W_{c ˙_{δ}}^{I, J} [ζ Φ]]), e, j \sum v_{e, j, δ} = t_{δ} ˙^{ζ} .

Here $ϑ_{e, j, δ}$ denotes the partition pulled back to the appropriate seed chart, including the fixed tip/away partition if used.

Individual $v_{e, j, δ}$ are not declared to be derivatives of raw sectors. No motion term is needed in (eq:postadjoint-partition). The different raw-sector convention is recorded in Appendix app:partitions.

Displayed formulas in this passage

Em02-A@6#eq:partition · line 637
$θ_{00, δ} θ_{0 R, δ} θ_{10, δ} θ_{01, δ} θ_{11, δ} = a_{1, δ} a_{2, δ} a_{R, δ}, = a_{1, δ} a_{2, δ} (1 - a_{R, δ}), = a_{1, δ} (1 - a_{2, δ}), = (1 - a_{1, δ}) a_{2, δ}, = (1 - a_{1, δ}) (1 - a_{2, δ}) .$
Em02-A@6#eq:postadjoint-partition · line 654
$v_{e, j, δ} (Φ) = - i I, J \sum I_{e} (ϑ_{e, j, δ} P_{e}^{I, J} [W_{c ˙_{δ}}^{I, J} [ζ Φ]]), e, j \sum v_{e, j, δ} = t_{δ} ˙^{ζ} .$
unlabeled · line 624
$θ_{tip} = b_{0} (∣ r ∣_{E}^{2} / ϵ_{0}^{2}), θ_{away} = 1 - θ_{tip},$
unlabeled · line 632
$a_{j, δ} = b_{0} (q_{j}^{2} / (L^{2} δ^{2})), j = 1, 2, a_{R, δ} = b_{0} (q_{R}^{2} / (L^{2} δ^{2})) .$