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## THE LAWN-ROTH THEOREM FOR POSSIBLE EVIDENCE OF S4 UNIVERSE

The Lawn-Roth theorem (Lawn-Roth-Embedding-Theorem) provides necessary and sufficient conditions for the embedding of a three-manifold into $S^4$ of sectional curvature $\kappa$ and therefore radius $1/\sqrt{\kappa}$:  the condition is the existence of two spinor fields $\phi_1,\phi_2$ satisfying $D\phi_1 = +(3/2 H + 3/2\sqrt{\kappa})\phi_1$ and $D\phi_2 = -(3/2 H + 3/2 \sqrt{\kappa})\phi_2$.  Now for the vacuum state of a compact three dimensional physical universe if there exist fermions with sufficiently small masses $2/3\sqrt{\kappa}$ then we can apply this Lawn-Roth theorem to conclude that it can be embedded isometrically into a four-sphere because in this case mean curvature $H=0$.  The mass gap problem is for bosons or Yang-Mills fields rather than fermions.