Hush
So you want a formal proof to completely destroy your bullshits huh?
Alright.
What if, the actual value of π is not transcendental?
What if, the actual value of π is greater than the approximated value?
What if, the actual circumference of the circle is inherently longer than the perimeter of the circumscribed n-gon at, and after a threshold n, and precisely because the circle has smooth curvature at all scale that the assumption of Archimedes where, the circumference remains shorter than the perimeter of circumscribed n-gon for all n fails to approximate the circumference, because it underestimates the circumference after the threshold n?
The actual value of π for such a scenario can now logically be postulated to be both algebraic and constructible, in complete opposing qualities with the transcendental area constant, ξ. Renamed it as ξ(pronounced as Xi) because this constant is rightfully NOT π. Instead, it is the constant derived strictly from the area of circle using the method of exhaustion, having the same value as modern approximated "pi".
From this postulate, we can add another postulate which serves as the final nail to the coffin. The actual value of π relates to the golden ratio, φ.
The range would be within ξ < π < 2√ξ, normalized using the square of equal area.
The ONLY value which fits all the postulates and within the range is 4÷√φ, ≈3.144605511029693144278234343371835718
or
≈3.1446
3.1446m is the value measured from the circumference of a 1m in diameter circle, by Harry Lear.
What makes 4÷√φ being the only correct value?
Within the construction of squaring the circumference(not squaring the area as that is already proven impossible as ξ is proven transcendental by Lindemann in year 1882), the intersection points are minimal while having maximal geometric symmetries. The construction is thus geometrically balanced and harmonious, adhering to the qualities of φ.
What's next?
The start of a spiritual evolution for the whole of humanity. Acceptance of the mathematical truth that a circle has two constants.
love || evol
If C > P for n ≥ n_threshold, π > ξ.
π is both algebraic and constructible.
π relates to the golden ratio, φ, and adheres to Optimality.
Geometric Proof:

Lukie Poole:
We need to check that it doesn't work for other polygons.
((sin(2ξ/n) - 0)/(-cos(2ξ/n) + (n/x)tan(ξ/n)))((sqrt(((n/x)tan(ξ/n))2 - ((x/n)cot(ξ/n))2))/(-(x/n)cot(ξ/n))) = -1
The formula is derived from perpendicular slopes. The slanted line which intersects the critical point, and its perpendicular line(not shown here).
For n=4, the value is 4÷√φ
For n≥5, the value is less than ξ and for n=∞, x is ξ. Rejected as x is not within the range of ξ < π < 2√ξ and, x cannot be ξ.
For n=3, it is within range but not constructible.
x = -(3 sqrt(3))/4 + 3/4 sqrt(-5 - 64/(143 + 27 sqrt(73))1/3 + 2 (143 + 27 sqrt(73))1/3) + 1/2 sqrt(-45/2 + 144/(143 + 27 sqrt(73))1/3 - 9/2 (143 + 27 sqrt(73))1/3 + 81/2 sqrt(3/(-5 - 64/(143 + 27 sqrt(73))1/3 + 2 (143 + 27 sqrt(73))1/3)))
Final step. Compare n=3 and n=4 then justify why n=4 is the correct value.
For n=3, the value is ≈98.47531231526% closer to ξ
For n=4, the value is ≈99.9% closer to ξ
We can now conclude that 4÷√φ is the correct value of π based on the justification that it fits all postulates and it is the closest value to ξ, being 99.9% closer, compared to the value for n=3, being 98.5% closer while being non-constructible as well.
100 - ((x-ξ)/ξ)×100