The binary critical section of the q,r,k-Nacci family has characteristic $x^k – 1$ polynomial $P_k(x) = x^k – 2x^{k-1} – 1$ for every $k ge 2$. Every recurrence solution therefore decomposes into a binary exponential term and a zero-mean $k$-periodic residual. We derive the exponential coefficient explicitly from the initial values and give a finite necessary-and-sufficient criterion for integer initial data to define a globally exponentially rounded additive sequence (GERAS). The order-three orbit beginning $1,3,6$ has coefficient $10/7$ and residual $(-3/7, 1/7, 2/7)$, so global exponential rounding holds with optimal error $3/7$. Its local doubling discrepancies are $1,0,-1$, excluding every fixed local QGS multiplier. The GERAS framework and this example are developed in the companion manuscript; the present contribution is the uniform binary-critical factorization and its family-specific initial-data criterion. The distinction between local and global rounding is retained throughout.