"Pi-drawer" - The ultimate extractor of the real π

Description

Abstract (English)

This amazing solution was attained through an inverse geometric process, as it allowed to exactly extrapolate the natural π from the circle itself, converting it into a manageable rectilinear segment. 

As a must, the introduction to the «Cosmogony of the Golden Section», which is closely related to π, being fundamental for the semantic and epistemological trait. 
NB - Its release date is prior to the demonstration of the actual "Pi-drawer".

This enhanced 2nd edition updates and replaces the previous one, not in terms of con­tent, which remains unchanged, but to offer additional considerations that aid understanding, after the laborious effort of structuring the initial conception.
For some minor typographical corrections of the first edition, it is integrated here afresh; to be completed by significant ac­ces­so­ry notions.

Methods

As to the theorem, both the basic one and here its alternative way always de­fin­ing the π as unknown - but instead of bringing the circle towards a square, it starts from the square to obtain a circle of equal perimeter - are e­ven­tu­al­ly re­placed by the closest analysis that setting the ¼π as a constant, re­leases the implicit proof into the Pi-drawer in an immediate and accessible way.

From the effect of that second perspective, new postulates emerge with im­pli­ca­tions, regarding the transition from the circle, whose monad is the root of the golden section, to the golden spiral of any density or amplitude.
Once discovered how the golden ratio pervades the nature of the circle, as its soul, here's a first look at how it amplifies its manifestations through the true Golden Spiral, which punctuates the squaring the circle itself with every ⅛ rotation.

Far from being a half circle with radius 1, [¼]π, which I like to call 'Platinum', is the pivot of the world.