Registered Proper-Arc Chains and Placement Fibers in a Circular Collapse System

Consider identity-placed singleton packets on an $n$-cycle, acted on by rotation $p$ and the adjacent idempotent $d:1\mapsto0$. Register an actual fusion that creates a prescribed proper source arc $F$ of length $r$, and retain its ordered parent-to-port origin, every later fusion involving its carrier, and the complete terminal placed partition. For $n\ge3$ and $2\le r<n$, we classify these generated summaries by a dependent bijection: an internal origin cut, a ternary code for the selected absorption chain, and an interval-cut and occupied-site parameter for the remaining packets. Every parameter is realized by one continuous raw $p,d$ execution. If the chain leaves $k$ atoms outside its carrier, its fixed-origin placement fiber has size $n\binom{n+k-2}{k}$. Summing the fibers gives an exact formula for the complete registered-summary set of every proper source arc. At $r=2$ and $F=\{0,1\}$ this recovers the fixed-pair classification. For each fixed $r$, the normalized diagonal coefficients and the unscaled count have algebraic generating functions, and the count is asymptotic to $\frac{4(r-1)}{3^{r+2}}n(9/2)^n$.