Defeating the PI without Pyramids summoning sounds impossible, but not when you have a crises in your Woodbook. To do so there are 2 different methods.

* METHOD 1 * 
This is an easy to implement method once you understand what you're doing. 
- Cut 3 ropes 3 times beside LakeCake
- Once you see the reflection of last cut fades, hit any Pole
- Repeat 3 times
- Once you see the reflection of last cut fades, open Woodbook 

* METHOD 2 *
This one is complicated, but CRISIS will last forever 
- Buy Items : 3x #101, #582, #588, 9x #032, #000 in ORDER
- Sell Everything else at any Mirrorless shop 
- Fight 3 VABs at any arena, 3 times 
- DROP ALL ITEMS after last VAB fight and open Woodbook
- Repeat 3 times 

Enjoy!