Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

First new calculation: conormals at a retained triple

J2: the retained characteristic configuration › First new calculation: conormals at a retained triple · lines 894–945

XIV-C@1#eq:triangle, XIV-C@1#eq:nus

For the $B_{6}$ incidence, the underlying scalar numerical factors are

E_{0}^{R} (x - u) F_{0} (y - u) F_{m} (y - x) .

Choose

ℓ = (1, 1, 0, 0), p = η (ℓ, \cdot) .

At the fixed triple in J0,

y_{0} - u_{0} = L ℓ, x_{0} - u_{0} = 2 L ℓ .

With the inherited time-ordered orientation, the wavefront selections at the line endpoints are

line E_{0}^{R} (x - u) F_{0} (y - u) F_{m} (y - x) endpoint covectors (- p, p) at (x, u) (p, - p) at (y, u) (- p, p) at (y, x) .

The retarded forward-null singularity permits both conormal signs. The time-ordered signs are those copied in XIII-C. A simultaneous reversal of Fourier convention reverses all three selections and leaves the calculation unchanged.

In the common vertex order $(x, u, y)$ these covectors are

ν_{R} = (- p, p, 0), ν_{1} = (0, - p, p), ν_{2} = (p, 0, - p) .

They are nonzero and

ν_{R} + ν_{1} + ν_{2} = 0.

The original weight $f (x_{0}) σ_{ε} (u_{0}) g (y_{0})$ is nonzero. None of the vertices coincides with another. This is a retained configuration of the frozen graph, not a point removed by collision exclusion.

This selection is the inherited characteristic cancellation. The new calculation below identifies its geometric rank and support consequences.

Displayed formulas in this passage

XIV-C@1#eq:triangle · line 898
$E_{0}^{R} (x - u) F_{0} (y - u) F_{m} (y - x) .$
XIV-C@1#eq:nus · line 927
$ν_{R} = (- p, p, 0), ν_{1} = (0, - p, p), ν_{2} = (p, 0, - p) .$
unlabeled · line 903
$ℓ = (1, 1, 0, 0), p = η (ℓ, \cdot) .$
unlabeled · line 907
$y_{0} - u_{0} = L ℓ, x_{0} - u_{0} = 2 L ℓ .$
unlabeled · line 913
$line E_{0}^{R} (x - u) F_{0} (y - u) F_{m} (y - x) endpoint covectors (- p, p) at (x, u) (p, - p) at (y, u) (- p, p) at (y, x) .$
unlabeled · line 934
$ν_{R} + ν_{1} + ν_{2} = 0.$