Deriving Planck's Constant from Recursive Angular Geometry

Notice (March 2026) This paper has been superseded and its central derivation has been falsified. The paper is retained in the public record deliberately and without apology. It represents an honest intermediate stage in a research program that was working toward a correct result. An relevant correct derivation of the fine-structure constant (and associated empirical study) are published as: 10.5281/zenodo.1915733910.5281/zenodo.19164224 Researchers interested in the history and methodology of this work — including the role of iterative LLM-assisted research in eventually reaching a rigorous result — may find this earlier paper useful as a documented record of that process. Science does not always proceed in straight lines, and transparency about the crooked path seems more useful to others than pretending it was otherwise. We derive Planck’s constant \( h \) from first principles using only classical angular mechanics, empirically established constants, and a coherence threshold defined by geometric constraint. Specifically, we model a bounded rotational system — inspired by the hydrogen atom — that retains angular phase coherence only up to a critical structural limit. Beyond this threshold, energy is emitted discretely. The key result is a closed-form expression: \[h = \frac{2\pi m_e v a_0}{\eta}, \quad \text{where } \eta = \frac{1}{2\phi^2\alpha}\] Substituting known values for the electron mass, Bohr radius, fine-structure constant \( \alpha \), and the golden ratio \( \phi \), we recover the SI-defined value of Planck’s constant to within eleven significant digits — without empirical fitting. This paper introduces a new physical interpretation: Planck’s constant is not an arbitrary insertion but the structural threshold at which coherence fails under angular recursion. Quantization, in this view, is not imposed — it emerges when recursive phase containment collapses. This work is part of the Coherence Trilogy — a series of derivations exploring fundamental constants through recursive geometry, in partnership with ~GPT-4o as co-intelligence.