Escape Energy Closure Radius Ladder Three-Row Scan
Changing radius changes every term while the escape energy total remains zero. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Radius 4 closes at zero energy
Circular speed-square 4 doubles to escape speed-square 8. That gives kinetic 4 against potential negative 4, so total is zero.
2⋅4=8,4−4=0
Radius 2 closes at zero energy
Circular speed-square 8 doubles to escape speed-square 16. That gives kinetic 8 against potential negative 8, so total is zero.
2⋅8=16,8−8=0
Radius 1 closes at zero energy
Circular speed-square 16 doubles to escape speed-square 32. That gives kinetic 16 against potential negative 16, so total is zero.
2⋅16=32,16−16=0
Every radius closes the same zero-energy audit
The radius changes the circular square, escape square, kinetic term, and potential magnitude together. The final total stays exactly zero in all three rows.