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

Theorem: Exact nonzero reference value

The three labels and the reference value › Restricted Einstein cubic · lines 1244–1290 · proof included or linked

II@2#thm:nonzero, II@2#eq:nonzero

At both $β_{+}$ and $β_{-}$ ,

W_{2} = - 6, W_{3} = W_{4} = 2,

and

m (β_{+}) = m (β_{-}) = 16/3.

Proof. Because the physical tensors have zero temporal entries, (eq:R) gives

R_{12} = - A, R_{13} = B, R_{14} = B .

The temporal momenta in $W_{2}$ are $2, - 1, - 1$ . Those in $W_{3}, W_{4}$ are $0, 1, - 1$ , up to permutation. Lemma lem:ABB therefore gives

W_{2} = - (4 + 1 + 1) = - 6, W_{3} = W_{4} = 2.

At $d_{*}$ , both species channels have $(D_{s}, D_{t}, D_{u}) = (0, 1, 1)$ . Equation (eq:m1122) gives

m = \frac{- 2 ( - 6 ) + 2 + 2}{3} = \frac{16}{3} .

The tree signs can also be displayed directly:

v_{4} (G, A, B, B) = - (W_{2} + W_{3} + W_{4}) = 2, (X_{2}, X_{3}, X_{4}) = (6, - 2, - 2) .

Thus

F (d_{*}) = \frac{2}{3} (2) - (0 \cdot 6 + 1 \cdot (- 2) + 1 \cdot (- 2)) = \frac{16}{3} .

Displayed formulas in this passage

II@2#eq:nonzero · line 1251
$m (β_{+}) = m (β_{-}) = 16/3.$
unlabeled · line 1247
$W_{2} = - 6, W_{3} = W_{4} = 2,$
unlabeled · line 1260
$R_{12} = - A, R_{13} = B, R_{14} = B .$
unlabeled · line 1268
$W_{2} = - (4 + 1 + 1) = - 6, W_{3} = W_{4} = 2.$
unlabeled · line 1274
$m = \frac{- 2 ( - 6 ) + 2 + 2}{3} = \frac{16}{3} .$
unlabeled · line 1279
$v_{4} (G, A, B, B) = - (W_{2} + W_{3} + W_{4}) = 2, (X_{2}, X_{3}, X_{4}) = (6, - 2, - 2) .$
unlabeled · line 1285
$F (d_{*}) = \frac{2}{3} (2) - (0 \cdot 6 + 1 \cdot (- 2) + 1 \cdot (- 2)) = \frac{16}{3} .$