Miller's Dimensional Scaling Theorem: The Tight Frame Identity as a Measurement Constraint

The tight frame identity in R^d states that for N equally-distributed unit vectors, the sum of squared projections onto any axis equals N/d. This paper proves the d=2 case, states the general result, and shows that the continuous limit and the power rule are different algebraic routes to the same ratio. The two parameters — N (modes present) and d (measurement dimensionality) — generate the structural constants across classical mechanics, quantum mechanics, thermodynamics, and cosmology: 1/2, 2/3, 1, 4/3, 3/2, 2, and 3. As applications, the Koide ratio Q=2/3 for charged lepton masses and the pion mass ratio m_π/m_p = 4/27 are both derived from the observable fraction d/N with zero free parameters. The universal cross-coupling P_× = 1/6 is derived as the observable fraction applied to adjacent modes at 120°, and underpins all electromagnetic coupling constants. The measurement dimensionality d=2 is not an assumption: wave persistence forces exactly two perpendicular components satisfying i²=-1, making d=2 geometrically necessary. The Koide application supersedes the separately published Phase Position Theorem.