Loops and the GCD:For all {$T\in\mathcal {B}^+$}, {$$\frac{N(\langle a,b,c\rangle)}2+2=\left\{\begin{array}{ll} x_{_T}+y_{_T}&\text{if } T \text{ is an axle}\\ x_{_T}+y_{_T}+1 &\text{if } T \text{ is a rim}\\ x_{_T}+y_{_T}-1&\text{if } T \text{ is generic or a spoke.}\end{array}\right.$$} Proof: Rims and the GCD: For all rims {$\langle a,b,c\rangle$}, {$\frac{N(\langle a,b,c\rangle)}2+2=x_{_T}+y_{_T}+1$}. Axles and the GCD: For all axles, {$\langle a,b,c\rangle$}, {$\frac{N(\langle a,b,c\rangle)}2+2=x_{_T}+y_{_T}$}. To prove the rest of Loops and the GCD we need to extend {$\Delta$} Respects Loops to boxes with negative height. Respecting Loops on Boxes with Negative Height: For all {$a,b\in\mathbb N^+$}, {$c\in\mathbb Z$}, {$c<0$}, {$a+b+c>0$}, {$N(\langle a,b,c\rangle )=N(\langle a,b,a+b+c\rangle )$}. Spokes and Generic Boxes and the GCD: For all spokes and generic boxes, {$\langle a,b,c\rangle$}, {$\frac{N(\langle a,b,c\rangle)}2+2=x_{_T}+y_{_T}-1$}. |