Euler-Alex Cylindrical Paradox

Euler-Alex Cylindrical Paradox. Suppose we have a big vertically
placed hollow cylindrical object with spiral engraving on the surface.
The Spiral paint a trajectory from a point at the top rim of this
object to another point at the bottom of this object. Now we place a
metal ball of a small size that could fit the width of this spiral on
the starting position of the spiral on the top of this object and
release it. Naturally, gravity would pull this object downward. Do the
Math now.
Now the potential energy is Mass * Height * g, assume the ball is
rotating with constant speed, the kinetic energy is centripetal Force
* Distance= Mass* Velocity^2 / Radius* Number of Circle from top to
bottom(n)* 2* Pi* Radius (We assume the Spiral is approximate equal to
the length of a circle, however, the perimeter of this ellipse should
be GREATER than a circle), which we could rewritten to 2* Mass *
Velocity^2* Pi* n. And in textbook Physics we assume all the kinetic
energy is coming from the Gravity, therefore:
Mass * Height * g=Mass * Velocity^2 *n *Pi or
Height* g= Mass* Velocity^2 * n* Pi
Are they TRULY equal each other? It is impossible Mathematically
since the value of n is arbitrary! Theoretically it could equal to any
amount we want it to be. Of course, the result still await experiment
verification.
My prediction is no amount of experiment would prove this equation
right since it is WRONG theoretically. The reason?
Our assumption is wrong because we omit something very important:
The attraction force between the cylindrical object and the metal ball
which PROVIDE the energy which transform an otherwise free fall into
circular motion. It doesn't matter what material the ball and the
cylindrical object are made of, as long as the spiral trajectory could
keep the ball rolling. We just use Gravitation Energy to Lead Out this
non-obvious energy. The amount of energy lead out depend on the
geometry of the trajectory, has nothing to do with the mass of the
rolling object.
To complete the Paradox, suppose we have a way to transfer the
kinetic energy into other energy, and storing that energy. Then we use
that energy to rise the ball from bottom to the top. However, the
amount of energy is more than adequate to do that. The energy input is
LESS than energy output!
To increase the level of challenge, we could change this rolling
ball into a Magnets, and place coil vertically inside the cylindrical
object. Use the electrical energy produce to move the magnetic from
destination to starting point. Since it is inconceivable that the
rolling Magnet would slow down, we have a very simple Generator which
extract energy from Magnetic Field, Molecular Attraction and
Gravitational Field at the same time. Now Output is far greater than
input.
To increase the level if challenge further, now we use what the
rolling Magnetic to somehow power a Generator of the design in 1. The
Output to Input ratio could be of ANY number we wanted to (greater
than 1).
We start with every working Physics theory to arrive at a result no
conventional Physicist would accept.
I challenge any Physicist to prove me wrong, or to nominate me for
the Nobel Prize on Physics for theorizing on Perpetual Motion Machine.