Quadrature of the Circle via Dimensionless Periodic Normalization

Version 6.

This work presents an axiomatic and fully dimensionless formulation of the quadrature of the circle as a consequence of periodic geometric normalization.

The construction is purely algebraic and is based on internal geometric relations rather than empirical measurement or physical calibration. Building on a previously established normalization identity (Ψ = 1), the framework shows that, under a normalized circular setting, the circumference of a unit-diameter circle assumes the value √10.

The result does not contradict classical impossibility results concerning ruler-and-compass constructions, as no classical geometric construction is claimed. Instead, the quadrature emerges within a normalized, dimensionless framework, where area–perimeter equivalence is defined through internal scaling relations.

All quantities introduced are dimensionless and symbolic. No physical interpretation or empirical claim is implied.