The 2πe Quantization Rule in exp(iθ): Discovery Through Extreme-Precision Computation

Authors

Radek M. Sobczynski, Chemled Technologies, LLCFollow

Abstract

ABSTRACT. We report a computational discovery: the Taylor‑series convergence of exp(iθ) exhibits a robust, period‑scaled quantization of term counts converging to 2πe per 2π cycle, realized as an alternating 17–18 term cadence. Using multiprecision arithmetic (up to 4.7×10⁶ decimal digits for 10⁶ periods) and Hull‑spiral visualization, we follow the term‑by‑term phasor sum and identify an “end burst” at the factorial–exponential crossover θⁿ ≈ n!, followed by a double‑exponential collapse of magnitude into a narrow post‑burst regime (the “Planckian abyss”). Unwrapped‑phase analysis reveals discrete phase bands intrinsic to the post‑burst geometry: in an 8‑period window the phase exhibits 26–27 quantized levels with characteristic 2–3 gap alternations, visible across multiple sampling resolutions. These results are reproducible across independent multiprecision implementations and are consistent with resurgence‑style structure in series tails. The refined combination 8*2πe+(7/17+7/18)/2 ≈ 137.03607436, emerging directly from the Hull‑spiral’s geometric constraints, shows striking proximity to 1/α, the fine‑structure constant. We present this as an empirical discovery requiring theoretical explanation; nevertheless, the observed precision indicates a profound connection between the geometry of Taylor series and fundamental constants.

Recommended Citation

Sobczynski, Radek M. (2025) "The 2πe Quantization Rule in exp(iθ): Discovery Through Extreme-Precision Computation," International Journal of Computer and Systems Engineering: Vol. 4: Iss. 2, Article 1.
Available at: https://digitalcommons.fairfield.edu/ijcase/vol4/iss2/1