Tilting cycle vs orbital cycle of earth Common sense tells us a year is conveniently divided exactly into four seasons. I'm wondering whether that means the earth's tilting cycle is equal to its orbital cycle. If so, there must be a reason. Do we have a proof?
There is no tilting cycle.

The Earth continues to tilt the same way all year round.  The axis points towards the pole star. But as the Earth goes round the sun, in December the northern hemisphere points away from the sun. In June the Northern hemisphere points towards the sun, as the Ascii art below shows (not to scale)

N                    N  
 \                    \
  O         *          O
   \                    \
    S                    S 
Earth                 Earth
in Dec     Sun       in Jun


So no special explanation is required. The Earth points the same way all year, it just moves relative to the sun.

The tilt of the Earth relative to the stars doesn't change. So the tilt of the Earth relative to the sun is entirely due to the Earth's orbit.  This proves that the cycle of tilting relative to the sun will take exactly one orbit to complete. It is no coincidence; the cycle of seasons will take exactly one year. 

(There is variation in the direction of tilt, called precession, but this is a much slower cycle, lasting 25000 years)