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.

This work is based on the Bouras 4D Root Method, which constitutes a proprietary and protected methodological framework.