regardless of that, it defies parelli's principle of "maintain direction" (true straightness) being priority 2, and his method of achieving that.
"maintain direction" can't be achieved by saying "left = more left". because the pattern will tell the horse that when it wants to go left, it should go more left.
maintain direction is achieved by "left = right" in equal measures. that way when the horse wants to go left, it will also go right in an equal amount, resulting in the horse staying straight.
the reason i teach the horse to go more in the opposing direction is because when the horse deviated it tells me the horse isn't even in the first place, so straightness no longer is the priority, evenness in left/right is the new priority. because evenness in left/right results in straightness.
i dont know how i can make this any simpler :s
i suppose an easy way of saying it is, if the horse were to make a 5 degree turn to the left, rider should from there take a 10 degree turn to the right. 5 degrees to get back to the straight, and another 5 degrees because straight lines aren't the issue, evenness in left/right is the issue, and the horse turning 5 to the left in the first place told us exactly how uneven the horse is, and exactly how much we have to do to solve it.
I find that explanation a bit easier to understand but feel that 5 and 10 degree changes are much different than "Circles".
In flying and navigation there is "course"
which is a straight line between two points and "heading"
which is the degrees that you point the plane to travel on the course line.
Horses are similar in that sometimes you have to make a heading change to stay on course and sometimes you just change your course.
Think of a stream and you want to get to the other side but the water is flowing very fast.
If you point the bow of the boat at the point that you want to reach the current will drag you down stream.
Your analogy has you over correcting to get back on course which will do that but I am not sure the horse understands what he did wrong.
Maybe we are saying the same thing but "True straightness" is a state of all things being balanced but there will always be very small corrections to maintain it.
I don't really feel that horses see lines like we do and really just want to get along with the least fuss.
To make a point I think you could lead a horse into the arena each day and line up feed pans in a perfectly straight line and let him walk from one to the other.
Each day I could spread the pans out further and further until there were only two pans (one on each end of the arena).
The horse would walk in a perfect line from one to the other because it would be the most efficient way to get there.
Now I could throw the horse off "Course" by sitting on him and leaning to the outside in one direction or another.
If I did the same thing every day for a year the horse would get wise to me throwing him off course and would compensate for my lack of balance and that is what most people are looking for in a horse.