One of the proofs of the Pythagorean Theorem in our high school geometry book was the single word BEHOLD! and three checkerboard-patterned rectangles forming a right triangle in the negative space they created. One checkerboard was a 3×3, one a 4×4, and one a 5×5; and, indeed, 9 plus 16 equals 25.
To prove the non-existence of Doodle Logic is impossible. No matter how random the doodle is, the doodler brings SOMETHING to the table, if the doodler is a human being. Any computer program will necessarily have code that imposes rules.
Perhaps our local Universe is the ultimate doodle.