Black Holes as Perpendicular Quaternion Projections: A First-Principles Derivation of Bekenstein–Hawking Entropy

A first-principles derivation of the Bekenstein–Hawking entropy coefficient 1/4 from the axiom that physical waves are unit-quaternion-valued. The average squared projection of a uniformly distributed point on the unit 3-sphere S³ in R⁴ onto any real axis is identically 1/4 by rotational symmetry — the n=4 instance of the same identity that gives ½mv² at n=2 and the radiation equation of state P = ρ/3 at n=3. Combined with a self-gravitation argument that fixes the Planck length as the minimum stable mode size (one quaternion mode per Planck-area pixel at any horizon), this gives S_BH = A/(4·ℓ_Pl²) with no fitted parameters. Black holes are interpreted as the geometric surfaces where one full quaternion component of every wave mode rotates perpendicular to the exterior observation frame; the 1/4 is the projection of the inaccessible perpendicular content. Hawking radiation emerges as parametric sideband generation by continuous Bogoliubov mixing between perpendicular and exterior wave components at the surface-gravity modulation frequency. The Smarr identity T_H · S_BH = Mc²/2, the Bekenstein entropy bound, and several mass-independent geometric horizon constants follow as corollaries. The construction predicts 1/4 is universal across all horizons in any spacetime dimension, in agreement with Wald's universal entropy formula and the Ryu–Takayanagi result that the same 1/4 appears for non-horizon minimal surfaces.