Rank-3 scale theorems: Difference between revisions

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 ====== MV3 implies LQ except in the case "xyzyx" (WIP) ======
-====== MV3 + LQ implies PMOS (WIP) ======
+====== MV3 + LQ implies EMOS (WIP) ======
+Proof sketch:
+Let L = L(t) = (a, b, c)t + (0, α, β) be a line in generic position corresponding to the signature aX bY cZ. The projection matrices
+<math>
+P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \
+P_2 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \
+P_3 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},
+</math>
+(which map (x, y, z) to (x, y), (y, z) and (z, x), respectively) map L to lines in R^2 that are in generic position (i.e. they intersect the x- and y-axes at irrational points). The projections record intersections with two of the planes to intersections with x- and y- axes, and these intersections must spell out the result of removing one of the step sizes; hence the resulting scales must be mosses.
+(MV3 has not been used yet)
+====== MV3 + EMOS implies PMOS (WIP) ======
 ====== PMOS implies AG (except in the case xyxzxyx) (WIP) ======