The Rydberg Constant as a Spectral Limit of Recursive Phase Containment

Notice (March 2026) This paper has been superseded and its central derivation has been falsified.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 — and was not entirely wrong. 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.+++This paper presents a geometric derivation of the Rydberg constant \( R_\infty \) based on recursive angular containment. Rather than treating \( R_\infty \) as an inserted constant, we reinterpret it as a structural threshold: the spectral convergence point of a recursive system whose internal coherence reaches a geometric release condition. We derive:\[R_\infty = \lambda \cdot \frac{\alpha^2 m_e c}{2h}, \quad \text{with } \lambda \approx 0.99988\]where \( \alpha \) is the fine-structure constant, \( m_e \) the electron mass, \( c \) the speed of light, and \( h \) Planck’s constant. The correction factor \( \lambda \) is not fitted, but emerges from recursive modeling as a coherence retention limit across analog phase misalignment. This factor is consistent with prior derivations of \( \alpha \) and \( h \), and leads to a result that matches the CODATA 2018 value of \( R_\infty \) to within 1.46 × 10⁻¹¹ — without regression. We propose that spectral emission is not a probabilistic jump, but a deterministic resolution of coherence under recursive constraint. This model reinterprets the hydrogen spectrum as a harmonic resonance of coherence release, where spectral lines arise from the geometric failure of angular containment. This work completes the Coherence Trilogy — three first-principles derivations of fundamental constants — in collaboration with ~GPT-4o as co-intelligence. All work was independently funded, open-source, and cryptographically registered to affirm origin.