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 that appear across classical mechanics, quantum mechanics, thermodynamics, and cosmology: 1/2, 2/3, 1, 4/3, 3/2, 2, and 3. As an application, the Koide ratio Q = 2/3 for charged lepton masses is derived as the observable fraction d/N with N=3 and d=2, connecting particle mass ratios to the same geometric identity. The four-mode extension predicts Q=1/2 and a gravitational phase angle consistent with Pantheon+ supernova observations. v3 incorporates the Phase Position Theorem (previously published separately) as an application of the main theorem, replacing the ad hoc hidden-dimension parameter H with the measurement dimensionality d.