Miller's Phase Position Theorem: Generalized Koide Ratio as a Dimensional Readout

The Koide ratio Q = Σmᵢ / (Σ√mᵢ)² = 2/3 for charged leptons has remained unexplained since 1982. We show that this ratio is an instance of a general formula: for N wave modes observed from a space with H hidden dimensions,

Q = (1 + H) / N

The charged lepton value Q = 2/3 corresponds to N = 3 modes with H = 1 hidden dimension. The individual masses encode the observer's phase position within the mode structure. When extended to four modes (N = 4, H = 1), the formula predicts Q = 1/2 and yields a fourth mass mG = 95.105 MeV that encodes a gravitational phase angle consistent with supernova observations (Pantheon+).

The theorem connects particle mass ratios to the dimensional structure of the space they inhabit: the ratio tells you how many dimensions exist and how many are hidden; the masses tell you where you stand.

This is the companion paper to Miller's Wave-Axis Theorem (doi:10.5281/zenodo.18884954).