Phi³ [calc. Windows] = 0,23606797749978969640917364616882
CD = (Phi³ /2) = 0,11803398874989484820458682308441
(Phi³ /2)² = 0,01393202250021030359082632860559
(0,25 -(Phi³ /2)² = 0,23606797749978969640917367139441
with which we can choose between two formulas:
via Golden
VA = AD × Φ
= 0,78615137775742328606955858071066
= 0,61803398874989484820458682629611
or via
Pythagorean theorem:
VA = AD²+VD²

= 0,78615137775742328606955857866799
= 0,61803398874989484820458682308441
= 0,61803398874989484820458681467588

Apart from insignificant differences in calculations beyond the 25th decimal, the square of a side VA² is equivalent to the linear measure VD, height of the tri­an­gle, which is Golden Section of the circle [diameter] as well; but what has po­lar­ized and concentrated all my resources in this work is the realization that VA is the real module, the tuning fork that makes all the strings of the math­e­mat­i­cal sys­tem vibrate in perfect and eloquent harmony, surpassing the fourth part of the π computed to date for only 0,00075.
While trying immediately to discover the underlying motivation, this leads us to believe and not without reason, or def­i­nite­ly means that the perimeter of the square with side AV, alias π [in the current project] is precisely:

3,1446055110296931442782343146719… !

and that therefore with VA supervenes – ideal lemma in this case – none other than the rectification virtual of a quarter of a circumference VQ.
Furthermore, since the length of its side represents in sq. the area of the circle with diameter mt.1, it will not be difficult to trace the square without the need for any calculation; although at this point the historic milestone appears to me as the least significant achievement of all, since:
we would have identified the π as Φ×4 !!!
that is, the essential and existential entity, which modulates in the cardinal di­rec­tions of space the ineffable detached balance, without gain or loss, of the e­ter­nal pre­sent, by virtue of the golden property on which it draws.

A magnificent fact, which informs the whole family of numbers and symbols in mathematics and geometry of new but eternal relationships between the at­a­vis­tic formulas, charging them with an unprecedented synergy in the sci­en­tif­ic path.

Suffice it to note that for the side of the large triangle, which in the circle with a di­am­e­ter of mt.1 is Φ, represents with mt. 0,78615 the side of the square pe­rim­e­ter, or ¼ of circumference, and with sqm. 0,78615 the area of the cir­cle. It means that according to these clarifications of a π geometric ie " native"

the circle maintains a special Golden Ratio with the circumscribed square!

An extraordinary chain of deductions derives from this, which scholars will not fail to expand (one of which is already in existence at the “Journal of Mod­ern Phys­ics” and reported below); but by now the primitive ratio of the circle with the square that contains it can be very illuminating, if not evidential: