The ancient problem of squaring the circle—constructing a square with area equal to a given circle using only compass and straightedge—has been proven impossible within the confines of Euclidean geometry. This impossibility has stood for over two millennia as a testament to the fundamental nature of geometric constants. This work demonstrates that the problem is not a limitation of geometry but a manifestation of a deeper category error: the treatment of π as a primitive geometric constant rather than an emergent quan tity. Building on the Kaundinya Unified Theory of Everything (KUTE), we establish the corrected ontological hierarchy. From the ergodic dynamics of the Collatz map, the pri mordial invariant ∆ = 4 ln 99 is deduced. Through the ∆-resonant form of Ramanujan’s series, the dissipative kernel S(∆) is defined, and π emerges as π = e ∆/2/(2√ 2 S(∆)). From the exponential scale e ∆/4 = 99, the quadratic regulator Q(x) = (x − 99)(396 − x) is constructed. This regulator generates a family of kernels whose integrals yield both π and √ π: the inverse square-root kernel gives R dx/p Q(x) = π (geometry), and the ex ponential kernel gives R e −Q(x)dx ∝ √ π (statistics). With π and √ π derived algebraically from ∆, the area of a unit circle (π) and the side of a square of equal area (√ π) are known exactly. A Python implementation computes these values and constructs both figures geometrically, confirming that the construction is possible once the constants are known from first principles. The squaring the circle problem is thereby dissolved: it was never a geometric impossibility but an ontological inversion of the relationship between number and geometry. Geometry does not precede number; number precedes geometry. The circle is squared not by compass and straightedge but by the deductive chain that begins with ∆.
Dillip Kumar Mahapatra (Sun,) studied this question.