Foundations of Tribonacci Quasigeometric Sequences

A Tribonacci quasigeometric sequence (TQGS) is an integer sequence constrained simultaneously by the exact third-order recurrence and nearest-integer multiplication by the Tribonacci constant τ. The framework introduced here is defined by $h_{n+3}=h_{n+2}+h_{n+1}+h_n$ , $h_{n+1}=\text{rnd}\left(\tau h_n\right)$ , ${\tau }^3={\tau }^2+\tau +1$ . Writing ${\delta }_n=h_{n+1}？\tau h_n$ gives the exact discrepancy propagation law ${\delta }_{n+2}=\left(1+\tau ？{\tau }^2\right){\delta }_n+\left(1？\tau \right){\delta }_{n+1}$ , $\left|{\delta }_n\right|<\frac{1}{2}$ .These relations establish the basic TQGS