Manuscript II@2 · II.tex · section · 50d64def

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Complete matrix encoding and the finite certificate › All raw entries · lines 1391–1438

II@2#eq:KH, II@2#eq:integer-encoding

At $β_{+}$ , the lower-index momenta and longitudinal tensor are

K = - 1 - 1 11 00 - 1 1 0000 1 - 1 00, H_{1} = 200 - 1 00000000 - 1 000 .

Put $H_{2} = A, H_{3} = H_{4} = B$ , $η_{a} = (- 1, 1, 1, 1)_{a}$ , and $J^{+} = (1, 1, 2, 2)$ .

For every original label $(I, P)$ , define

c_{I, P} = - σ \in S_{4} I_{r} = J_{σ (r)}^{+} (r = 1, \dots, 4) \sum λ_{1}, \dots, λ_{10} = 0 \sum^{3} K_{σ (1), λ_{1}} K_{σ (2), λ_{2}} \times r = 1 \prod^{4} (H_{σ (r)})_{λ_{2 r + 1} λ_{2 r + 2}} {u, v} \in P \prod η_{λ_{u}} δ_{λ_{u}, λ_{v}} .

The first index of $K$ labels legs $1, \dots, 4$ ; the second labels components $0, \dots, 3$ . All inputs and terms are integers.

Let

I = (3 - I_{1}, 3 - I_{2}, 3 - I_{3}, 3 - I_{4}) .

Displayed formulas in this passage

II@2#eq:KH · line 1394
$K = - 1 - 1 11 00 - 1 1 0000 1 - 1 00, H_{1} = 200 - 1 00000000 - 1 000 .$
II@2#eq:integer-encoding · line 1416
$c_{I, P} = - σ \in S_{4} I_{r} = J_{σ (r)}^{+} (r = 1, \dots, 4) \sum λ_{1}, \dots, λ_{10} = 0 \sum^{3} K_{σ (1), λ_{1}} K_{σ (2), λ_{2}} \times r = 1 \prod^{4} (H_{σ (r)})_{λ_{2 r + 1} λ_{2 r + 2}} {u, v} \in P \prod η_{λ_{u}} δ_{λ_{u}, λ_{v}} .$
unlabeled · line 1435
$I = (3 - I_{1}, 3 - I_{2}, 3 - I_{3}, 3 - I_{4}) .$