"Pi-drawer" - The ultimate extractor of the real π – the final theorem

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.

Methods (English)

Following a period of essential maturation, this third edition of the Pi-drawer retraces the conceptual framework that led to the immediate extraction of π from the circle.
It explains the underlying premises and the steps of its development in a direct, straightforward manner, aiming to maximize understanding of a procedure that stands on its own — almost as if it needed no demonstration, but only a clear vision of objective data.

It may seem like an oxymoron, but its greatest challenge for me was precisely its implicit, elusive circular simplicity. As for me, forgive my lack of academic formalism, I belong to an antiquity in which the primary qualification was knowledge.
Be that as it may, the Pi-drawer is the only method available to mathematical science throughout the millennia known to current civilization, for calculating the native Pi rather than an inevitably approximate simulation of it, given the geometric principle whereby an arc —even one consisting of just three imaginary points— will always be longer than the chord formed by its two endpoints. .

As things stand, is the prevailing value of π 3.14159 sufficiently reliable for all types of calculations?

The Pi-drawer proves the opposite, since if one pours such fervor into trillions of decimal places, the third digit of the correct 3.14460 counts more than all the subsequent ones. 
It does so in two stages, through a process that — in spite of no small amount of effort — I might present as elementary:

1. it maps one-eighth of the circumference onto a line segment, thus identifying it geometrically.
2. generates a clear and well-founded comparison context, which will allow it to be calculated
   with impeccable algebraic precision, without the need for speculation.

It is simply a matter of acknowledging this — something that, I believe, the entire scientific community will not be enabled to put off for too many Pi Days.

Technical info

Even though each statement presupposes a unitary Cartesian circle, my final summary of the theorem, –framed on the last page– may appear a bit ambiguous, being referred to a radius = 1, but with the conclusion that it is applicable ('always') to circles of any size.

I am slightly rephrasing it for clarity in version 3.1, to describe the radius =1 as represented by CB, so that ¼π is to be understood as multiplied ×CB where CB is ≠ 1.

In order not to reduce the immediate perception of the Pi-drawer formula, I only specify its general expression below.

Other

GIVEN A RIGHT-ANGLED TRIANGLE IN WHICH THE HYPOTENUSE IS EQUAL TO THE DIVISION OF THE TWO LEGS, MULTIPLIED BY THE GREATER, THE SMALLER SIDE WILL ALWAYS BE EQUAL TO ONE-EIGHTH OF THE CIRCUMFERENCE HAVING THE GREATER LEG AS ITS RADIUS.
From this, we deduce the real π .