Miller's Wave-Axis Theorem: The Projection Ratio of Wave-Based Measurement

A wave is a two-dimensional object — it oscillates perpendicular to its propagation axis. Projecting it onto the one-dimensional axis gives a ratio of 2:1. We prove the geometric identity that captures this: for N ≥ 3 equally-spaced unit vectors in the plane, the sum of squared projections onto any axis equals N/2, yielding an observable fraction of 2/N (the tight frame condition). We then show that the continuous limit — the time-average ⟨cos²(ωt)⟩ = 1/2 — and the power rule ∫v dv = v²/2 that produces the factor of 1/2 in the work–energy theorem are different algebraic routes to the same dimensional ratio. The mathematical content is elementary. The observation that these are instances of a single geometric fact — 2D projected onto 1D — and the interpretation of that fact as a measurement constraint in wave-based systems, is new.