Per-Period Rotation-Closure Theorem: A 17|18-Term Invariant for Euler Rotations

Authors

• Radek M. Sobczynski, Chemled Technologies, LLCFollow

Abstract

This paper provides the formal analytic foundation for the 2πe Quantization Rule first reported in “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. In that work, the 2πe threshold emerged empirically as the minimum number of Taylor terms required to preserve rotational closure of exp(iθ) over one period. Here we supply the complementary analytic derivation, establishing the Per‑Period Rotation‑Closure Theorem: a finite‑order structural invariant showing that uniform closure of the unit‑circle trajectory requires a truncation order exceeding a Stirling‑derived floor with leading constant 2πe. Rounded to integers, this yields the invariant 17|18‑term requirement per rotation. The proof uses Stirling’s approximation with remainder to show that the highest retained term must be uniformly suppressed across the interval, thereby eliminating the possibility that the observed quantization was a numerical artifact. The result is a closed‑form theorem of rotational resolution, confirming that the 2πe constant is a genuine structural invariant of finite‑order Euler rotations and not a computational anomaly.

Recommended Citation

Sobczynski, Radek M. (2026) "Per-Period Rotation-Closure Theorem: A 17|18-Term Invariant for Euler Rotations," International Journal of Computer and Systems Engineering: Vol. 5: Iss. 1, Article 1.
Available at: https://digitalcommons.fairfield.edu/ijcase/vol5/iss1/1