A K₄ Framework for π: Rotational Symmetry, the Geometric Degree of Freedom, and a Unified Research Program

We identify a K₄ structure in the mathematics of π: the four classical objects carrying π — the Euclidean disk, the Gaussian distribution, the Gamma function, and Euler's formula — are connected by six named mathematical relationships that appear to form a complete graph, all simultaneously indexed by GDoF (d) = ⌊d/2⌋, the Geometric Degree of Freedom. No specialist in any of the four domains — geometry, probability, analysis, algebra — would find their own domain's π-indexed table surprising. What is surprising, and what has not previously been identified, is that all four tables share exactly the same index in every dimension with no exception, connected to each other by six classical relationships every specialist already knows. This paper names that index, proves the geometric base case, and asks the question the pattern forces: why? The primary technical contribution is Theorem 3 (Shared-Rotation Variance Cancellation): structured rotational sampling using an n-coreset eliminates all non-resonant Fourier modes of the support function of a convex body via the roots-of-unity identity, achieving deterministic variance collapse at rate O (n^−2s) for Cˢ bodies — a rate improvement of n^2s−1/2 over independent Monte Carlo sampling. The geometric base case is Theorem 0: for any centrally symmetric convex body B ⊂ ℝ², rotation averaging under Haar measure on O (2) produces a Euclidean disk with circumference-to-diameter ratio π, independent of B. The GDoF Equipartition Conjecture proposes that each GDoF contributes one factor of π to every K₄ object. The Gaussian integral ∫䅜㵧 e^−|x|² dx = π^d/2 is identified as the apparent master object in which all four vertices meet. Three candidate additional vertices are identified: number theory (Riemann zeta function at even integers), topology (Chern-Gauss-Bonnet theorem), and information theory (maximum entropy characterization of the Gaussian). Whether the full graph is K₄, K₇, or infinite is identified as the deepest open question raised by this paper. Applied consequences in high-dimensional statistics and signal processing follow from Theorem 3 directly. The formal proof of each K₄ edge and the capstone conjecture — that geometric constants are Haar measure normalizations of symmetry groups in the Erlangen sense — constitutes the six-direction research program.