This paper applies the Temporal Dynamics Framework to the Banach-Tarski paradox. The Banach-Tarski rotation angle encodes Universal Prime 3 and connects to the fine structure constant. The trigonometric relationship sin(137 degrees) approximately equals 0.682, matching the Planck 2018 dark energy fraction of 68.3%. The ratio cos-squared(137 degrees)/2 approximately equals 0.267, matching dark matter at 26.8%, with baryonic matter at approximately 5.1%. The value 1/137 is interpreted as the observation threshold between measurable and non-measurable mathematical sets. All 18 verification tests pass. Paper 8 in the Temporal Dynamics Framework Research Series.
James Norman Ibbotson (Mon,) studied this question.