Noncrystallographic tail-triangle C-groups of rank 4 and interlacing number 2

This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and $k = 3, 4, 5,$ or $6$. As moduli, we use the primes in the quadratic integer ring $\mathbb{Z}[\tau]$, where $\tau = \frac{1 + \sqrt{5}}{2}$, the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable.